Method of designing thickness of coating film and semiconductor photonic device

ABSTRACT

A coating film is provided on an end surface of a semiconductor photonic element including an active layer through which light propagates. The coating film has a two-layer structure including a first layer film and a second layer film arranged in a stacked relation. The thicknesses of the first and second layer films are determined so that the value of the amplitude reflectivity of the coating film is equal to an imaginary number.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a semiconductor photonic deviceincluding a semiconductor laser device for use as a light source foroptical information processing, a signal source for opticalcommunication and a pumping source for a fiber amplifier, a lightemitting diode device, a semiconductor amplifier device, a semiconductormodulator and the like, and to a coating film for use in thesemiconductor photonic device.

2. Description of the Background Art

In general, a coating film is formed on an end surface of asemiconductor photonic element in a semiconductor photonic device suchas a semiconductor laser device and a light emitting diode device forpurposes of protection and reflectivity adjustment. Conventionally,various studies have been done on the power reflectivity of such acoating film.

A technique for forming an anti-reflection coating film on an endsurface of a semiconductor laser element is disclosed, for example, in:K. Shigihara et al., “Antireflection coating for laser diodes,”ELECTRONICS LETTERS, 31st Aug. 1995, Vol. 31, No. 18, pp. 1574-1576; andJapanese Patent Application Laid-Open No. 5-243689 (1993). According tothis technique, if a single-layer coating film having a refractive indexn₁ and a thickness d₁ is formed on an end surface of a semiconductorlaser having an effective refractive index n_(c), the coating filmbecomes an anti-reflection film when the following conditions aresatisfied: n₁=n_(c) ^(1/2) and d₁=λ₀/(4·n₁), where λ₀ is the value ofthe lasing wavelength of the semiconductor laser. At this time, themagnitude of the reflection amplitude vector of the coating film iszero. The “reflection amplitude vector” used herein refers to theamplitude vector of a reflected wave obtained when the amplitude vectorof an incident wave (referred to hereinafter as an “incident amplitudevector”) is placed on the positive real axis in a complex plane and isdefined to have a magnitude of “1.” Thus, the reflection amplitudevector is a vector indicative of the position of an amplitudereflectivity in the complex plane.

A technique for improving the design flexibility of an anti-reflectioncoating film is disclosed in Japanese Patent Application Laid-Open No.2004-88049.

A coating film having a power reflectivity greater than zero isdisclosed in Japanese Patent Application Laid-Open No. 2004-289108.According to Japanese Patent Application Laid-Open No. 2004-289108, ifthe coating film is a single-layer film, the power reflectivity of thecoating film takes on a relative minimum value not equal to zero whenthe following conditions are satisfied: n₁≠n_(c) ^(1/2) andd₁=λ₀/(4·n₁)×m (where m is an odd number). Thus, when n₁≠n_(c) ^(1/2)and the thickness of the coating film is an odd multiple of λ₀/(4·n₁),the reflection amplitude vector of the coating film is present on thenegative real axis in a complex plane. In this case, there is a phasedifference of 180 degrees between the incident amplitude vector and thereflection amplitude vector. Then, the imaginary components of theincident and reflection amplitude vectors are equal to zero, and adifference between the real components of the vectors is equal to apower transmissivity. The value of the power reflectivity is obtained bysubtracting the power transmissivity from one.

For designing the coating film having a power reflectivity greater thanzero, the conventional techniques establish the condition that thethickness of the coating film is an odd multiple of λ₀/(4·n₁) asdescribed above to result in the reflection amplitude vector positionedon the real axis in the complex plane. It is hence necessary todetermine the thickness of the coating film so that the amplitudereflectivity is a real number when designing the thickness of thecoating film having the power reflectivity greater than zero. Thisresults in a low degree of design flexibility of the thickness of thecoating film, and creates a likelihood that the coating film having adesired characteristic is not designed.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide a technique capableof improving the design flexibility of the thickness of a coating filmprovided on an end surface of a semiconductor photonic element.

A first aspect of the present invention is intended for a method ofdesigning the thickness of a coating film including a plurality oflayers and provided on an end surface of a semiconductor photonicelement including an active layer through which light propagates.According to the present invention, the method includes the followingsteps (a) and (b). The step (a) is to select an imaginary number as avalue of an amplitude reflectivity of the coating film. The step (b) isto determine the thickness of each of the plurality of layers of thecoating film so that the value of the amplitude reflectivity of thecoating film is equal to the imaginary number selected in the step (a).

According to a second aspect of the present invention, a semiconductorphotonic device includes a semiconductor photonic element, and a coatingfilm. The semiconductor photonic element, includes an active layerthrough which light propagates. The coating film includes a plurality oflayers and is provided on an end surface of the semiconductor photonicelement. The coating film has an amplitude reflectivity taking on avalue set at an imaginary value.

The use of the imaginary value as the value of the amplitudereflectivity of the coating film makes it possible to design thethickness of the coating film having a predetermined power reflectivityin consideration for more complex numbers having the same amplitude asthe value of the amplitude reflectivity than real numbers when used.This improves the design flexibility of the thickness of the coatingfilm to make the coating film having a desired characteristic easy todesign.

These and other objects, features, aspects and advantages of the presentinvention will become more apparent from the following detaileddescription of the present invention when taken in conjunction with theaccompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows the amplitude reflectivity of a coating film represented ina complex plane;

FIG. 2 is a side view showing a structure of a semiconductor photonicdevice having a single-layer coating film on an end surface of asemiconductor photonic element;

FIG. 3 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 4 is a side view showing a structure of a semiconductor photonicdevice according to the present invention;

FIG. 5 is a flowchart showing a method of designing the thickness of acoating film according to the present invention;

FIG. 6 is a side view showing a structure of the semiconductor photonicdevice according to a first preferred embodiment of the presentinvention;

FIG. 7 is a graph showing the wavelength dependence of the powerreflectivity of a coating film according to the first preferredembodiment of the present invention;

FIG. 8 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 9 is a graph showing the wavelength dependence of the powerreflectivity of the coating film according to the first preferredembodiment of the present invention;

FIG. 10 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 11 is a graph showing the wavelength dependence of the powerreflectivity of a coating film according to a second preferredembodiment of the present invention;

FIG. 12 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 13 is a graph showing the wavelength dependence of the powerreflectivity of the coating film according to the second preferredembodiment of the present invention;

FIG. 14 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 15 is a side view showing a structure of the semiconductor photonicdevice according to a third preferred embodiment of the presentinvention;

FIG. 16 is a graph showing the wavelength dependence of the powerreflectivity of a coating film according to the third preferredembodiment of the present invention;

FIG. 17 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 18 is a graph showing the wavelength dependence of the powerreflectivity of the coating film according to the third preferredembodiment of the present invention;

FIG. 19 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 20 is a side view showing a structure of the semiconductor photonicdevice according to a fourth preferred embodiment of the presentinvention;

FIG. 21 is a graph showing the wavelength dependence of the powerreflectivity of a coating film according to the fourth preferredembodiment of the present invention;

FIG. 22 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 23 is a graph showing the wavelength dependence of the powerreflectivity of the coating film according to the fourth preferredembodiment of the present invention;

FIG. 24 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 25 is a side view showing a structure of the semiconductor photonicdevice according to a fifth preferred embodiment of the presentinvention;

FIG. 26 is a graph showing the wavelength dependence of the powerreflectivity of a coating film according to the fifth preferredembodiment of the present invention;

FIG. 27 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 28 is a graph showing the wavelength dependence of the powerreflectivity of the coating film according to the fifth preferredembodiment of the present invention;

FIG. 29 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 30 is a side view showing a structure of the semiconductor photonicdevice according to a sixth preferred embodiment of the presentinvention;

FIG. 31 is a graph showing the wavelength dependence of the powerreflectivity of a coating film according to the sixth preferredembodiment of the present invention;

FIG. 32 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 33 is a graph showing the wavelength dependence of the powerreflectivity of the coating film according to the sixth preferredembodiment of the present invention;

FIG. 34 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 35 is a side view showing a structure of the semiconductor photonicdevice according to a seventh preferred embodiment of the presentinvention;

FIG. 36 is a graph showing the wavelength dependence of the powerreflectivity of a coating film according to the seventh preferredembodiment of the present invention;

FIG. 37 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 38 is a graph showing the wavelength dependence of the powerreflectivity of the coating film according to the seventh preferredembodiment of the present invention;

FIG. 39 is a side view showing a structure of the semiconductor photonicdevice according to an eighth preferred embodiment of the presentinvention;

FIG. 40 is a graph showing the wavelength dependence of the powerreflectivity of a coating film according to the eighth preferredembodiment of the present invention;

FIG. 41 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 42 is a graph showing the wavelength dependence of the powerreflectivity of the coating film according to the eighth preferredembodiment of the present invention;

FIG. 43 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 44 is a graph showing the wavelength dependence of the powerreflectivity of a coating film according to a ninth preferred embodimentof the present invention;

FIG. 45 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 46 is a graph showing the wavelength dependence of the powerreflectivity of the coating film according to the ninth preferredembodiment of the present invention;

FIG. 47 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 48 is a graph showing the wavelength dependence of the powerreflectivity of a coating film according to a tenth preferred embodimentof the present inventions;

FIG. 49 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 50 is a graph showing the wavelength dependence of the powerreflectivity of the coating film according to the tenth preferredembodiment of the present invention;

FIG. 51 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 52 is a graph showing the wavelength dependence of the powerreflectivity of a coating film according to an eleventh preferredembodiment of the present invention;

FIG. 53 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 54 is a graph showing the wavelength dependence of the powerreflectivity of the coating film according to the eleventh preferredembodiment of the present invention;

FIG. 55 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 56 is a graph showing the wavelength dependence of the powerreflectivity of a coating film according to a twelfth preferredembodiment of the present invention;

FIG. 57 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 58 is a graph showing the wavelength dependence of the powerreflectivity of the coating film according to the twelfth preferredembodiment of the present invention;

FIG. 59 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 60 is a graph showing the wavelength dependence of the powerreflectivity of a coating film according to a thirteenth preferredembodiment of the present invention;

FIG. 61 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 62 is a graph showing the wavelength dependence of the powerreflectivity of the coating film according to the thirteenth preferredembodiment of the present invention;

FIG. 63 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 64 is a graph showing the wavelength dependence of the powerreflectivity of a coating film according to a fourteenth preferredembodiment of the present invention;

FIG. 65 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 66 is a graph showing the wavelength dependence of the powerreflectivity of the coating film according to the fourteenth preferredembodiment of the present invention;

FIG. 67 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 68 is a graph showing the wavelength dependence of the powerreflectivity of a coating film according to a fifteenth preferredembodiment of the present invention;

FIG. 69 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 70 is a graph showing the wavelength dependence of the powerreflectivity of the coating film according to the fifteenth preferredembodiment of the present invention;

FIG. 71 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 72 is a graph showing the wavelength dependence of the powerreflectivity of a coating film according to a sixteenth preferredembodiment of the present invention;

FIG. 73 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 74 is a graph showing the wavelength dependence of the powerreflectivity of the coating film according to the sixteenth preferredembodiment of the present invention;

FIG. 75 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 76 is a graph showing the wavelength dependence of the powerreflectivity of a coating film according to a seventeenth preferredembodiment of the present invention;

FIG. 77 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 78 is a graph showing the wavelength dependence of the powerreflectivity of the coating film according to the seventeenth preferredembodiment of the present invention;

FIG. 79 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 80 is a graph showing the wavelength dependence of the powerreflectivity of a coating film according to an eighteenth preferredembodiment of the present invention;

FIG. 81 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 82 is a graph showing the wavelength dependence of the powerreflectivity of the coating film according to the eighteenth preferredembodiment of the present invention;

FIG. 83 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 84 is a graph showing the wavelength dependence of the powerreflectivity of a coating film according to a nineteenth preferredembodiment of the present invention;

FIG. 85 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 86 is a graph showing the wavelength dependence of the powerreflectivity of the coating film according to the nineteenth preferredembodiment of the present invention;

FIG. 87 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 88 is a graph showing the wavelength dependence of the powerreflectivity of a coating film according to a twentieth preferredembodiment of the present invention;

FIG. 89 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film;

FIG. 90 is a graph showing the wavelength dependence of the powerreflectivity of the coating film according to the twentieth preferredembodiment of the present invention;

FIG. 91 is a graph showing the wavelength dependence of the powerreflectivity of the single-layer coating film; and

FIGS. 92 and 93 are tables listing conditions and results according tothe first to twentieth preferred embodiments of the present invention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

FIG. 1 shows that the amplitude reflectivity r of a coating filmprovided on an end surface of a semiconductor photonic element having anactive layer, such as a semiconductor laser element, is represented in acomplex plane. A phase angle θ in FIG. 1 is an angle formed by apositive real axis and a reflection amplitude vector rv, that is, avector indicative of the position of the amplitude reflectivity r in thecomplex plane.

When the wavelength of light propagating through the active layer isdenoted generally by λ, the amplitude reflectivity r for the light isexpressed by r=r_(r)(λ)+ir_(i)(λ) where r_(r)(λ) and r_(i)(λ) are thereal part and the imaginary part, respectively, of the amplitudereflectivity r, i is an imaginary unit, and i²=−1.

For λ=λ₀, the amplitude reflectivity r equals zero and a powerreflectivity R expressed by |r|² also equals zero when r_(r)(λ) andr_(i)(λ) satisfyr _(r)(λ₀)=0  (1)r _(i)(λ₀)=0  (2)

Thus, an anti-reflection coating film is prepared by designing thecoating film so that Equations (1) and (2) are satisfied.

The above-mentioned method, however, cannot be employed for thepreparation of a coating film having a power reflectivity R greater thanzero, rather than the anti-reflection coating film. FIG. 2 is a sideview showing a structure of a semiconductor photonic device having asingle-layer coating film 2 provided on an end surface 1 b of asemiconductor photonic element 1 including an active layer 1 a. Thepower reflectivity R of the single-layer coating film 2 in contact withfree space 3 filled with air or nitrogen as shown in FIG. 2 takes on arelative minimum value for the wavelength λ=λ₀ when the followingequation is satisfied: $\begin{matrix}{d_{f} = {\frac{\lambda_{0}}{4n_{f}}\left( {{2m} + 1} \right)}} & (3)\end{matrix}$where d_(f) is the thickness of the single-layer coating film 2, n_(f)is the refractive index of the coating film 2, and m is a non-negativeinteger such as 0, 1 and 2. The amplitude reflectivity r at this time isexpressed by: $\begin{matrix}{r = \frac{n_{c} - n_{f}^{2}}{n_{c} + n_{f}^{2}}} & (4)\end{matrix}$where n_(c) is the effective refractive index of the semiconductorphotonic element 1.

Setting the thickness d_(f) of the coating film 2 so that Equation (3)is satisfied causes the power reflectivity R to take on a relativeminimum value of 4% for the wavelength λ=λ₀ when the effectiverefractive index n_(c) of the semiconductor photonic element 1 is 3.37and the refractive index n_(f) of the coating film 2 is 2.248 or 1.499.

FIG. 3 is a graph showing the wavelength dependence of the powerreflectivity R of the coating film 2 when λ₀=980 nm, n_(c)=3.37 andn_(f)=1.499 are set. In FIG. 3, a curve 103 is plotted for the thicknessd_(f)=λ₀/(4 n_(f))=169.08 nm of the coating film 2, and a curve 104 isplotted for the thickness d_(f)=5 λ₀/(4 n_(f))=845.41 nm.

As illustrated in FIG. 3, the power reflectivity R takes on a relativeminimum value of 4% for the wavelength λ of 980 nm, whether thethickness d_(f) is set at λ₀/(4 n_(f)) or at 5 λ₀/(4 n_(f)). For thethickness d_(f) set at λ₀/(4 n_(f)), the curve 103 shows that awavelength band for which the power reflectivity R falls within ±2% fromthe relative minimum value of 4% is from 848 nm to 1161 nm to provide awavelength bandwidth of 313 nm. For the thickness d_(f) set at 5 λ₀/(4n_(f)), the curve 104 shows that a wavelength band for which the powerreflectivity R falls within ±2% from the relative minimum value of 4% isfrom 951 nm to 1011 nm to provide a wavelength bandwidth of 60 nm. Thevalue obtained by dividing the wavelength bandwidth of 60 nm by 981 nmthat is the median value of the wavelength band is approximately 0.06,which is a measure of the extent of the wavelength band. For the coatingfilm 2 as shown in FIG. 2, as the thickness d_(f) thereof is increasedin increments of an odd multiple of λ₀/(4 n_(f)), the wavelength bandfor which the power reflectivity R falls within a predetermined rangebecomes narrower.

As discussed above, when the thickness d_(f) of the coating film 2 isset at an odd multiple of λ₀/(4 n_(f)), the amplitude reflectivity r isa real number as will be understood from Equation (4) described above,and thus the reflection amplitude vector rv for the coating film 2 ispresent on the real axis. In other words, it is necessary to design thethickness of the coating film so that the reflection amplitude vector rvis positioned on the real axis. Thus, this method is low in the degreeof design flexibility of the coating film, and sometimes cannot providea desired characteristic.

The present invention employs an imaginary number, i.e. a complex numberhaving a nonzero imaginary part, as the value of the amplitudereflectivity r to make it possible to design the thickness of thecoating film having a predetermined power reflectivity R inconsideration for various complex numbers having the same amplitude asthe value of the amplitude reflectivity, thereby improving the designflexibility of the thickness of the coating film. The basic principle ofthe present invention will now be described by using the coating filmhaving a two-layer structure as an example.

Basic Principle of Present Invention

FIG. 4 is a side view showing a structure of a semiconductor photonicdevice provided with a coating film 6 having a two-layer structure onthe end surface 1 b of the semiconductor photonic element 1. Thesemiconductor photonic device shown in FIG. 4 may be used as asemiconductor laser device, a light emitting diode device, asemiconductor amplifier device or a semiconductor modulator. As shown inFIG. 4, the semiconductor photonic element 1 has the active layer 1 awhich propagates light therethrough. When the semiconductor photonicelement 1 is a semiconductor laser element, the light generated in theactive layer 1 a is repeatedly reflected from a pair of cladding layers(not shown) which hold the active layer 1 a therebetween, whereby laserlight is outputted from the semiconductor photonic element 1.

The coating film 6 includes a first layer film 4 having a refractiveindex n₁ and a thickness d₁, and a second layer film 5 having arefractive index n₂ and a thickness d₂. The first layer film 4 and thesecond layer film 5 are stacked in the order named on the end surface 1b of the semiconductor photonic element 1 including an end surface ofthe active layer 1 a.

When the wavelength of light propagating through the active layer 1 a isdenoted generally by λ, the amount of phase change φ₁ of light in thefirst layer film 4 and the amount of phase change φ₂ of light in thesecond layer film 5 are expressed respectively as: $\begin{matrix}{\phi_{1} = {\frac{2\pi}{\lambda}n_{1}d_{1}}} & \left( {5a} \right) \\{\phi_{2} = {\frac{2\pi}{\lambda}n_{2}d_{2}}} & \left( {5b} \right)\end{matrix}$

The amplitude reflectivity r in the complex plane is expressed as:$\begin{matrix}{r = \frac{{\left( {m_{11} + m_{12}} \right)n_{c}} - \left( {m_{21} + m_{22}} \right)}{{\left( {m_{11} + m_{12}} \right)n_{c}} + \left( {m_{21} + m_{22}} \right)}} & (6)\end{matrix}$where m₁₁, m₁₂, m₂₁ and m₂₂ are elements of a characteristic matrix forthe coating film 6, and satisfy the following determinant:$\begin{matrix}{\begin{bmatrix}m_{11} & m_{12} \\m_{21} & m_{22}\end{bmatrix} = {\begin{bmatrix}{\cos\quad\phi_{1}} & {{- \frac{\mathbb{i}}{n_{1}}}\sin\quad\phi_{1}} \\{{- {\mathbb{i}}}\quad n_{1}\sin\quad\phi_{1}} & {\cos\quad\phi_{1}}\end{bmatrix} \times \begin{bmatrix}{\cos\quad\phi_{2}} & {{- \frac{\mathbb{i}}{n_{2}}}\sin\quad\phi_{2}} \\{{- {\mathbb{i}}}\quad n_{2}\sin\quad\phi_{2}} & {\cos\quad\phi_{2}}\end{bmatrix}}} & \left( {7a} \right)\end{matrix}$

Using Equation (7a), Equation (6) is rewritten as: $\begin{matrix}{r = \frac{\begin{matrix}{{\left( {n_{c} - 1} \right)\cos\quad\phi_{1}\cos\quad\phi_{2}} + {\left( {\frac{n_{1}}{n_{2}} - \frac{n_{2}n_{c}}{n_{1}}} \right)\sin\quad\phi_{1}\sin\quad\phi_{2}} - {\mathbb{i}}} \\\left\{ {{\left( {\frac{n_{c}}{n_{2}} - n_{2}} \right)\cos\quad\phi_{1}\sin\quad\phi_{2}} + {\left( {\frac{n_{c}}{n_{1}} - n_{1}} \right)\sin\quad\phi_{1}\cos\quad\phi_{2}}} \right\}\end{matrix}}{\begin{matrix}{{\left( {n_{c} + 1} \right)\cos\quad\phi_{1}\cos\quad\phi_{2}} - {\left( {\frac{n_{2}n_{c}}{n_{1}} + \frac{n_{1}}{n_{2}}} \right)\sin\quad\phi_{1}\sin\quad\phi_{2}} - {\mathbb{i}}} \\\left\{ {{\left( {\frac{n_{c}}{n_{2}} + n_{2}} \right)\cos\quad\phi_{1}\sin\quad\phi_{2}} + {\left( {\frac{n_{c}}{n_{1}} + n_{1}} \right)\sin\quad\phi_{1}\cos\quad\phi_{2}}} \right\}\end{matrix}}} & \left( {6a} \right)\end{matrix}$

Equation (6a) defines the amplitude reflectivity r by using therefractive indices n₁ and n₂ and the amounts of phase change φ₁ and φ₂for the respective layers of the coating film 6, and the effectiverefractive index n_(c) of the semiconductor photonic element 1. Theamount of phase change φ₁ is defined by the thickness d₁, the refractiveindex n₁ and the wavelength λ of the light propagating through theactive layer 1 a according to Equation (5a). The amount of phase changeφ₂ is defined by the thickness d₂, the refractive index n₂ and thewavelength λ according to Equation (5b). Therefore, it may be said thatEquation (6) defines the amplitude reflectivity r based on therefractive indices n₁ and n₂, the effective refractive index n_(c), thethicknesses d₁ and d₂, and the wavelength λ.

Because the magnitude |r| of the amplitude reflectivity r of the coatingfilm 6 having the power reflectivity R=R_(t) is equal to R_(t) ^(1/2),the amplitude reflectivity r is positioned on a circle with its centerat the origin and with a radius equal to R_(t) ^(1/2). Thus, fordesigning the coating film 6 having the power reflectivity R=R_(t) whereR_(t) is the design value of the power reflectivity R, it is necessaryto set the position of the amplitude reflectivity r in the complex planat any position lying on the circle with its center at the origin andwith the radius equal to R_(t) ^(1/2). In other words, it is necessaryto define the reflection amplitude vector rv as any vector with itsinitial point at the origin and with a magnitude equal to R_(t) ^(1/2).The design value R_(t) of the power reflectivity R is referred tohereinafter as a “design reflectivity R_(t).”

In accordance with the above, the coating film 6 having the powerreflectivity R=R_(t) is designed by determining the amounts of phasechange φ₁ and φ₂ so that the value of the amplitude reflectivity r inEquation (6a) equals a complex number positioned at any point lying onthe circle with its center at the origin and with the radius equal toR_(t) ^(1/2) and then determining the thicknesses d₁ and d₂ of therespective layers of the coating film 6 using Equations (5a) and (5b). Amethod of determining the thicknesses d₁ and d₂ of the respective layersof the coating film 6 having the power reflectivity R=R_(t) for thewavelength λ=λ_(t) will be described below.

FIG. 5 is a flowchart showing a method of designing the thickness of acoating film. As shown in FIG. 5, any single point lying on a circlewith its center at the origin and with a radius equal to R_(t) ^(1/2) isselected in a complex plane in Step s1. In Step s2, a complex numberpositioned at the point selected in Step s1 is substituted as the valueof the amplitude reflectivity r into Equation (6a). In Step s3, therefractive indices n₁ and n₂ of the respectively layers of the coatingfilm 6 and the effective refractive index n_(c) are substituted intoEquation (6a). This provides Equation (6a) in which the amounts of phasechange φ₁ and φ₂ are unknowns.

Next, the left-hand and right-hand sides of Equation (6a) obtained inStep s3 are decomposed into a real part and an imaginary part, and anequation for the real part and an equation for the imaginary part areformulated in Step s4. If the values on the left-hand and right-handsides of Equation (6a) are, for example, a+ib and c+id respectively, theequation a=b and the equation c=d are formulated. This provides a systemof two simultaneous equations with two unknowns. In Step s5, the valuesof the amounts of phase change φ₁ and φ₂ are found from the system oftwo simultaneous equations obtained in Step s4.

In Step s6, the thicknesses d₁ and d₂ are found by substituting theamounts of phase changes φ₁ and φ₂ found in Step s5 into Equations (5a)and (5b) respectively, and further substituting the design value λ_(t)of the wavelength λ into Equations (5a) and (5b). This determines thethicknesses d₁ and d₂ of the respective layers of the coating film 6.The design value λ_(t) of the wavelength λ is referred to hereinafter asa “design wavelength λ_(t).”

Thus, the coating film 6 which satisfies the power reflectivity R=R_(t)for the wavelength λ=λ_(t) is designed by substituting the complexnumber having the magnitude of R_(t) ^(1/2) as the value of theamplitude reflectivity r into Equation (6a) and then determining thethicknesses d₁ and d₂ using the equations obtained by the substitution.

If a real number is employed as the value of the amplitude reflectivityr as disclosed in Japanese Patent Application Laid-Open No. 2004-289108described above, only a point on the positive real axis or a point onthe negative real axis is selectable in Step s1. As a result, there areonly two pairs of thicknesses d₁ and d₂ determined in Step s6. It ishence impossible to ensure a sufficient degree of design flexibility ofthe thickness of the coating film 6.

To solve such a problem, the present invention employs an imaginarynumber as the value of the amplitude reflectivity r for substitutioninto Equation (6a) to determine the thickness of each of the layers ofthe coating film 2 so that the value of the amplitude reflectivity r isan imaginary number. This allows the selection of any point on thecircle with its center at the origin and with the radius equal to R_(t)^(1/2) except the points on the real axis in Step s1, thereby to achievethe selection of more than two points. Thus, more complex numbers havingthe same magnitude can be substituted into Equation (6a) in Step s2, andmore pairs of thicknesses d₁ and d₂ can be determined in Step s6. Thisimproves the design flexibility of the thickness of the coating film 6to make the coating film 6 having a desired characteristic easy todesign.

First to twentieth preferred embodiments to be described belowillustrate the characteristics of coating films designed using a filmthickness designing method according to the present invention undervarious conditions established regarding a layer structure of thecoating films and the like.

First Preferred Embodiment

FIG. 6 is a side view showing a structure of the semiconductor photonicdevice provided with a coating film 13 having a six-layer structure onthe end surface 1 b of the semiconductor photonic element 1. Asillustrated in FIG. 6, the coating film 13 according to the firstpreferred embodiment of the present invention includes: a first layerfilm 7 having the refractive index n₁ and a thickness Ad₁; a secondlayer film 8 having the refractive index n₂ and a thickness Ad₂; a thirdlayer film 9 having the refractive index n, and a thickness Bd₁; afourth layer film 10 having the refractive index n₂ and a thickness Bd₂;a fifth layer film 11 having the refractive index n₁ and a thicknessCd₁; and a sixth layer film 12 having the refractive index n₂ and athickness Cd₂.

Each of the first layer film 7, the third layer film 9 and the fifthlayer film 11 is a tantalum oxide (Ta₂O₅) layer, and each of the secondlayer film 8, the fourth layer film 10 and the sixth layer film 12 is asilicon oxide (SiO₂) layer. Thus, the coating film 13 according to thefirst preferred embodiment is composed of two material layers: thetantalum oxide layer and the silicon oxide layer.

In the first preferred embodiment, the first and second layer films 7and 8, the third and fourth layer films 9 and 10, and the fifth andsixth layer films 11 and 12 constitute unit layer pairs each composed ofthe tantalum oxide layer and the silicon oxide layer arranged in astacked relation. In other words, the coating film 13 according to thefirst preferred embodiment includes three unit layer pairs each composedof the tantalum oxide layer and the silicon oxide layer arranged in astacked relation. The reference characters A, B and C which determinethe thicknesses of the respective layers of the coating film 13designate parameters individually determined for the respective unitlayer pairs and each indicating a contribution ratio of the thickness ofa corresponding unit layer pair to the thickness of the entire coatingfilm 13. The reference characters d₁ and d₂ according to the firstpreferred embodiment designate basic thicknesses individually determinedfor the respective material layers. Thus, the first preferred embodimentdetermines the thicknesses of the respective layers of the coating film13 by multiplying the basic thicknesses by the parameters indicating thecontribution ratios.

The amounts of phase change for the first to sixth layer films 7 to 12are designated by Aφ₁, Aφ₂, Bφ₁, Bφ₂, Cφ₁ and Cφ₂, respectively, usingEquations (5a) and (5b). The reference characters φ₁ and φ₂ according tothe first preferred embodiment designate the basic amounts of phasechange individually determined for the respective material layers.Therefore, the elements m₁₁, m₁₂, m₂₁ and m₂₂ of a characteristic matrixfor the coating film 13 according to the first preferred embodimentsatisfy the following determinant: $\begin{matrix}\begin{matrix}{\begin{bmatrix}m_{11} & m_{12} \\m_{21} & m_{22}\end{bmatrix} = {{\begin{bmatrix}{\cos\quad A\quad\phi_{1}} & {{- \frac{\mathbb{i}}{n_{1}}}\sin\quad A\quad\phi_{1}} \\{{- {\mathbb{i}}}\quad n_{1}\sin\quad A\quad\phi_{1}} & {\cos\quad A\quad\phi_{1}}\end{bmatrix}\begin{bmatrix}{\cos\quad A\quad\phi_{2}} & {{- \frac{\mathbb{i}}{n_{2}}}\sin\quad A\quad\phi_{2}} \\{{- {\mathbb{i}n}_{2}}\sin\quad A\quad\phi_{2}} & {\cos\quad A\quad\phi_{2}}\end{bmatrix}} \times}} \\{{\begin{bmatrix}{\cos\quad B\quad\phi_{1}} & {{- \frac{\mathbb{i}}{n_{1}}}\sin\quad B\quad\phi_{1}} \\{{- {\mathbb{i}}}\quad n_{1}\sin\quad B\quad\phi_{1}} & {\cos\quad B\quad\phi_{1}}\end{bmatrix}\begin{bmatrix}{\cos\quad B\quad\phi_{2}} & {{- \frac{\mathbb{i}}{n_{2}}}\sin\quad B\quad\phi_{2}} \\{{- {\mathbb{i}}}\quad n_{2}\sin\quad B\quad\phi_{2}} & {\cos\quad B\quad\phi_{2}}\end{bmatrix}} \times} \\{\begin{bmatrix}{\cos\quad C\quad\phi_{1}} & {{- \frac{\mathbb{i}}{n_{1}}}\sin\quad C\quad\phi_{1}} \\{{- {\mathbb{i}}}\quad n_{1}\sin\quad C\quad\phi_{1}} & {\cos\quad C\quad\phi_{1}}\end{bmatrix}\begin{bmatrix}{\cos\quad C\quad\phi_{2}} & {{- \frac{\mathbb{i}}{n_{2`}}}\sin\quad C\quad\phi_{2}} \\{{- {\mathbb{i}}}\quad n_{2}\sin\quad C\quad\phi_{2}} & {\cos\quad C\quad\phi_{2}}\end{bmatrix}}\end{matrix} & \left( {7b} \right)\end{matrix}$

For designing the thickness of the coating film 13 so that the value ofthe amplitude reflectivity r is an imaginary number, the first preferredembodiment previously determines the values of the parameters A to C,and executes Steps s1 to s5 using Equations (6) and (7b) described aboveto determine the values of the basic amounts of phase change φ₁ and φ₂.Thereafter, Step s6 is executed to determine the values of the basicthicknesses d₁ and d₂, and the thicknesses of the respective layers ofthe coating film 13 are determined using the previously determinedvalues of the parameters A to C and the values of the basic thicknessesd₁ and d₂ determined in Step s6. If the designed characteristic of thecoating film 13 is insufficient, the basic thicknesses d₁ and d₂ aredetermined again by changing the parameters A to C, and the thicknessesof the respective layers of the coating film 13 are designed again.

In the first preferred embodiment, the effective refractive index n_(c)of the semiconductor photonic element 1 is “3.37.” The refractive indexn₁ is “2.057” which is greater than n_(c) ^(1/2) (=1.84), and therefractive index n₂ is “1.480” which is less than n_(c) ^(1/2). Thedesign wavelength λ_(t) is 980 nm.

For designing the thickness of the coating film 13 having the powerreflectivity R of 4% (R_(t)=4%) when the wavelength λ equals the designwavelength of 980 nm under the above-mentioned conditions, a point whichprovides a phase angle θ of 60 degrees is selected, for example, in Steps1 so that the reflection amplitude vector rv is located in the firstquadrant (or the upper right quadrant) of the complex plane. Then,because the magnitude of the complex number at the selected point is0.2, the values of the real and imaginary parts r_(r) and r_(i) of thecomplex number inputted as the value of the amplitude reflectivity r are“+0.1” and “+0.17320508,” respectively, in Step s2.

When A=1.22, B=1.84 and C=2.19 are set, the basic amounts of phasechange φ₁ and φ₂ determined in Step s5 are “1.3449” and “0.463002,”respectively. Accordingly, the thicknesses of the first to sixth layerfilms 7 to 12 determined in Step s6 are 124.41 nm, 59.53 nm, 187.64 nm,89.78 nm, 223.33 nm and 106.86 nm, respectively.

FIG. 7 shows the wavelength dependence of the power reflectivity R ofthe coating film 13 thus designed, that is, how the power reflectivity Rchanges as the wavelength λ is hypothetically changed. As illustrated inFIG. 7, the power reflectivity R is 4% when the wavelength λ equals thedesign wavelength of 980 nm. A wavelength band for which the powerreflectivity R is approximately equal to the design reflectivity of 4%is wide. The power reflectivity R falls within a range from 3.3% to 6.0%for the wavelength λ ranging from 833 nm to 1078 nm.

When the allowable range of the power reflectivity R (referred tohereinafter as an “allowable reflectivity range”) is, for example, ±2%from the design reflectivity of 4%, the wavelength band for which thepower reflectivity R falls within the allowable reflectivity range isalso from 833 nm to 1078 nm, to provide a wavelength bandwidth W of 245nm. The center wavelength λ_(c) of the wavelength band is 956 nm. Thevalue obtained by dividing the wavelength bandwidth W by the centerwavelength λ_(c) is approximately 0.256, which is a measure of theextent of the wavelength band. This value is greater than 0.06, and isalso sufficiently greater than that obtained when the above-mentionedcoating film 2 having the characteristic indicated by the curve 103 ofFIG. 3 is provided on the end surface 1 b of the semiconductor photonicelement 1. Therefore, it may be said that the wavelength band for whichthe power reflectivity R falls within the allowable reflectivity rangeis a wide band.

If the value of the center wavelength λ_(c) has a decimal part, thevalue of the center wavelength λ_(c) is herein rounded to the nearestwhole number. The same is true for a value t_(r) to be described laterwhich is a quarter of the center wavelength λ_(c).

Because the thicknesses of the first to sixth layer films 7 to 12 takeon the above-mentioned values, the optical thickness t of the coatingfilm 13, i.e. the sum of the products of the refractive indices andthicknesses of the respective layers of the coating film 13, is 1480.41nm. This value is approximately 6.19 times the value t_(r) (239 nm)which is a quarter of the center wavelength λ_(c), and is sufficientlygreater than 3 λ_(c)/4. Thus, the coating film 13 is a very thick film.This improves heat dissipation characteristics at the end surface 1 b ofthe semiconductor photonic element 1 to suppress the increase intemperature of the end surface 1 b.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison with the semiconductor photonic device of the first preferredembodiment. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R takes on a relative minimum value of 4% for thewavelength λ equal to λ_(c) (956 nm) when the coating film 2 having arefractive index n_(f) of “1.4989” and a thickness d_(f) of five times159.45 nm, i.e. five times λ_(c)/(4n_(f)), is provided on the endsurface 1 b of the semiconductor photonic element 1. FIG. 8 shows thewavelength dependence of the power reflectivity R in this case. Asillustrated in FIG. 8, the wavelength band for which the powerreflectivity R falls within ±2% from the design reflectivity of 4% isfrom 928 nm to 986 nm, to provide a wavelength bandwidth W_(r) of 58 nm.

The above-mentioned value “1.4989” of the refractive index n_(f) isobtained by substituting R_(t)=0.04 and n_(c)=3.37 intoR _(t)=((n _(c) −n _(f) ²)/(n _(c) +n _(f) ²))²  (8)

As described above, the wavelength bandwidth W (245 nm) for the coatingfilm 13 of the first preferred embodiment is greater than the wavelengthbandwidth W_(r) (58 nm) for the coating film 2 shown in FIG. 2. It maybe said that the wavelength band for the coating film 13 is a wide band.

In general, the wavelength λ of light propagating through the activelayer 1 a is sometimes varied from the design wavelength λ_(t) dependingon temperature change and the like. To obtain stable characteristics ofthe semiconductor photonic device even if the wavelength λ is varied, itis desirable that the center wavelength λ_(c) of the wavelength band forwhich the power reflectivity R falls within the allowable reflectivityrange be equal to or close to the design wavelength λ_(t). For example,the center wavelength λ_(c) takes on a value close to the designwavelength λ_(t) of 980 nm when the thickness of the coating film 13 isdetermined using the basic thicknesses d₁ and d₂ obtained by setting thebasic amounts of phase change φ₁ and φ₂ at “1.3449” and “0.463002,”respectively, in a similar manner to the above instance and substituting1006 nm, rather than the design wavelength λ_(t), for λ in Equations(5a) and (5b). FIG. 9 shows the wavelength dependence of the powerreflectivity R in this case.

As illustrated in FIG. 9, the power reflectivity R falls within a rangefrom 3.3% to 6.0% for the wavelength λ ranging from 855 nm to 1107 nm.The wavelength band for which the power reflectivity R falls within ±2%from the design reflectivity of 4% is also from 855 nm to 1107 nm, toprovide a wavelength bandwidth W of 252 nm. The center wavelength λ_(c)of the wavelength band is 981 nm, which is very close to the designwavelength of 980 nm. The value obtained by dividing the wavelengthbandwidth W (252 nm) by the center wavelength λ_(c) (981 nm) isapproximately 0.257, which is greater than 0.06 and shows that thewavelength band for which the power reflectivity R falls within theallowable reflectivity range is a wide band.

In this case, the thicknesses of the first to sixth layer films 7 to 12of the coating film 13 are 127.71 nm, 61.11 nm, 192.63 nm, 92.16 nm,229.26 nm and 109.69 nm, respectively. The optical thickness t of thecoating film 13, i.e. the sum of the products of the refractive indicesand thicknesses of the respective layers of the coating film 13, is1519.69 nm. This value is approximately 6.20 times the value t_(r) (245nm) which is a quarter of the center wavelength λ_(c). Thus, the coatingfilm 13 is a very thick film. This improves heat dissipationcharacteristics at the end surface 1 b of the semiconductor photonicelement 1 to suppress the increase in temperature of the end surface 1b.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison with the semiconductor photonic device including the coatingfilm 13 having the characteristics shown in FIG. 9. For thesemiconductor photonic device shown in FIG. 2, the power reflectivity Rtakes on a relative minimum value of 4% for the wavelength λ equal toφ_(c) (981 nm) when the coating film 2 having the above-mentionedrefractive index n_(f) of “1.4989” and a thickness d_(f) of five times163.62 nm, i.e. five times λ_(c)/(4n_(f)), is provided on the endsurface 1 b of the semiconductor photonic element 1. FIG. 10 shows thewavelength dependence of the power reflectivity R in this case. Asillustrated in FIG. 10, the wavelength band for which the powerreflectivity R falls within ±2% from the design reflectivity of 4% isfrom 951 nm to 1011 nm, to provide a wavelength bandwidth W_(r) of 60nm.

Thus, even when the center wavelength λ_(c) is made closer to the designwavelength λ_(t), the wavelength bandwidth W (252 nm) for the coatingfilm 13 is greater than the wavelength bandwidth W_(r) (60 nm) for thecoating film 2 shown in FIG. 2.

Although the center wavelength λ_(c) is close to the design wavelengthλ_(t) in the above instance, the center wavelength λ_(c) may be madeexactly equal to the design wavelength λ_(t) by adjusting the values ofthe parameters A to C or the value substituted for λ in Equations (5a)and (5b).

If the coating film 13 is absent on the end surface 1 b of thesemiconductor photonic element 1, the value of the power reflectivity Ron the end surface 1 b is approximately 29.4%. This value is defined bythe effective refractive index n_(c) (3.37) of the semiconductorphotonic element 1 and a refractive index in the free space 3, and iscalculated as ((3.37−1)/(3.37+1))² assuming that the refractive index inthe free space 3 is “1.”

As illustrated in FIG. 7, the power reflectivity R is 4% when thewavelength λ is equal to the design wavelength λ_(t). Therefore, whenthe wavelength λ takes on the same value as the design wavelength λ_(t),the power reflectivity R for the semiconductor photonic device of thefirst preferred embodiment is lower than the power reflectivity R on theend surface 1 b obtained when the coating film 13 is absent on the endsurface 1 b of the semiconductor photonic element 1.

Second Preferred Embodiment

FIG. 11 shows the wavelength dependence of the power reflectivity R ofthe coating film 13 in the semiconductor photonic device according tothe second preferred embodiment of the present invention. Thesemiconductor photonic device according to the second preferredembodiment is similar to the semiconductor photonic device according tothe first preferred embodiment except that an alumina (Al₂O₃) layer isemployed in place of the silicon oxide layer as the material layer forthe second, fourth and sixth layer films 8, 10 and 12 of the coatingfilm 13. Thus, the coating film 13 according to the second preferredembodiment is composed of two material layers: the tantalum oxide layerand the alumina layer.

Because the alumina layer is employed in place of the silicon oxidelayer, the refractive index n₂ according to the second preferredembodiment is “1.620.” The design wavelength λ₁ is set at 808 nmaccording to the second preferred embodiment.

For designing the thickness of the coating film 13 having the powerreflectivity R of 4% (R_(t)=4%) when the wavelength λ equals the designwavelength of 808 nm in the semiconductor photonic device as mentionedabove, a point which provides a phase angle θ of 150 degrees isselected, for example, in Step s1 so that the reflection amplitudevector rv is located in the second quadrant (or the upper left quadrant)of the complex plane. Then, the values of the real and imaginary partsr_(r) and r_(i) of the complex number inputted as the value of theamplitude reflectivity r are “−0.17320508” and “+0.1,” respectively, inStep s2.

When A=2.15, B=2.00 and C=2.00 are set, the basic amounts of phasechange φ₁ and φ₂ determined in Step s5 are “0.587068” and “1.04832,”respectively. Accordingly, the thicknesses of the first to sixth layerfilms 7 to 12 determined in Step s6 are 78.91 nm, 178.92 nm, 73.40 nm,166.43 nm, 73.40 nm and 166.43 nm, respectively.

FIG. 11 shows the wavelength dependence of the power reflectivity R ofthe coating film 13 thus designed. As illustrated in FIG. 11, the powerreflectivity R is 4% when the wavelength λ equals the design wavelengthof 808 nm. The wavelength band for which the power reflectivity R isapproximately equal to the design reflectivity of 4% is wide. The powerreflectivity R falls within a range from 3.6% to 6.0% for the wavelengthλ ranging from 779 nm to 962 nm.

When the allowable reflectivity range is, for example, ±2% from thedesign reflectivity of 4%, the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range is alsofrom 779 nm to 962 nm, to provide a wavelength bandwidth W of 183 nm.The center wavelength λ_(c) of the wavelength band is 871 nm. The valueobtained by dividing the wavelength bandwidth W by the center wavelengthλ_(c) is approximately 0.210, which is greater than 0.06. It may be saidthat the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range is a wide band.

Because the thicknesses of the first to sixth layer films 7 to 12 takeon the above-mentioned values, the optical thickness t of the coatingfilm 13, i.e. the sum of the products of the refractive indices andthicknesses of the respective layers of the coating film 13, is 1293.37nm. This value is approximately 5.93 times the value t_(r) (218 nm)which is a quarter of the center wavelength λ_(c), and is sufficientlygreater than 3 λ_(c)/4. Thus, the coating film 13 is a very thick film.This improves heat dissipation characteristics at the end surface 1 b ofthe semiconductor photonic element 1 to suppress the increase intemperature of the end surface 1 b.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R takes on a relative minimum value of 4% for thewavelength λ equal to λ_(c) (871 nm) when the coating film 2 having therefractive index n_(f) of “1.4989” as in the first preferred embodimentand a thickness d_(f) of five times 145.27 nm, i.e. five timesλ_(c)/(4n_(f)), is provided on the end surface 1 b of the semiconductorphotonic element 1. FIG. 12 shows the wavelength dependence of the powerreflectivity R in this case. As illustrated in FIG. 12, the wavelengthband for which the power reflectivity R falls within ±2% from the designreflectivity of 4% is from 845 nm to 899 nm, to provide a wavelengthbandwidth W_(r) of 54 nm.

Thus, the wavelength bandwidth W (183 nm) for the coating film 13 of thesecond preferred embodiment is greater than the wavelength bandwidthW_(r) (54 nm) for the coating film 2 shown in FIG. 2.

As mentioned above, because the wavelength λ of light propagatingthrough the active layer 1 a is sometimes varied, it is desirable thatthe center wavelength λ_(c) of the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range be equal toor close to the design wavelength λ_(t). For example, the centerwavelength λ_(c) takes on a value close to the design wavelength λ_(t)of 808 nm when the thickness of the coating film 13 is determined usingthe basic thicknesses d₁ and d₂ obtained by setting the basic amounts ofphase change φ₁ and φ₂ at “0.587068” and “1.04832,” respectively, in asimilar manner to the above instance and substituting 751 nm, ratherthan the design wavelength λ_(t), for λ in Equations (5a) and (5b). FIG.13 shows the wavelength dependence of the power reflectivity R in thiscase.

As illustrated in FIG. 13, the power reflectivity R falls within a rangefrom 3.3% to 6.0% for the wavelength λ ranging from 724 nm to 894 nm.The wavelength band for which the power reflectivity R falls within ±2%from the design reflectivity of 4% is also from 724 nm to 894 nm, toprovide a wavelength bandwidth W of 170 nm. The center wavelength λ_(c)of the wavelength band is 809 nm, which is very close to the designwavelength of 808 nm. The value obtained by dividing the wavelengthbandwidth W (170 nm) by the center wavelength λ_(c) (809 nm) isapproximately 0.210, which is greater than 0.06 and shows that thewavelength band for which the power reflectivity R falls within theallowable reflectivity range is a wide band.

In this case, the thicknesses of the first to sixth layer films 7 to 12of the coating film 13 are 73.34 nm, 166.29 nm, 68.23 nm, 154.69 nm,68.23 nm and 154.69 nm, respectively. The optical thickness t of thecoating film 13, i.e. the sum of the products of the refractive indicesand thicknesses of the respective layers of the coating film 13, is1202.14 nm. This value is approximately 5.95 times the value t_(r) (202nm) which is a quarter of the center wavelength λ_(c). Thus, the coatingfilm 13 is a very thick film.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 4% for the wavelength λ equal to λ_(c) (809 nm)when the coating film 2 having the above-mentioned refractive indexn_(f) of “1.4989” and a thickness d_(f) of five times 134.93 nm, i.e.five times λ_(c)/(4n_(f)), is provided on the end surface 1 b of thesemiconductor photonic element 1. FIG. 14 shows the wavelengthdependence of the power reflectivity R in this case. As illustrated inFIG. 14, the wavelength band for which the power reflectivity R fallswithin ±2% from the design reflectivity of 4% is from 784 nm to 834 nm,to provide a wavelength bandwidth W_(r) of 50 nm.

Thus, even when the center wavelength λ_(c) is made closer to the designwavelength λ_(t), the wavelength bandwidth W (170 nm) for the coatingfilm 13 is greater than the wavelength bandwidth W_(r) (50 nm) for thecoating film 2 shown in FIG. 2.

Although the center wavelength λ_(c) is close to the design wavelengthλ_(t) in the above instance, the center wavelength λ_(c) may be madeexactly equal to the design wavelength λ_(t) by adjusting the values ofthe parameters A to C or the value substituted for λ in Equations (5a)and (5b), as in the first preferred embodiment.

Third Preferred Embodiment

FIG. 15 is a side view showing a structure of the semiconductor photonicdevice according to the third preferred embodiment of the presentinvention. A coating film 21 having a seven-layer structure is providedon the end surface 1 b of the semiconductor photonic element 1 in thethird preferred embodiment.

As illustrated in FIG. 15, the coating film 21 according to the thirdpreferred embodiment of the present invention includes: a first layerfilm 14 having the refractive index n₂ and a thickness Od₂; a secondlayer film 15 having the refractive index n₁ and the thickness Ad₁; athird layer film 16 having the refractive index n₂ and the thicknessAd₂; a fourth layer film 17 having the refractive index n₁ and thethickness Bd₁; a fifth layer film 18 having the refractive index n₂ andthe thickness Bd₂; a sixth layer film 19 having the refractive index n₁and the thickness Cd₁; and a seventh layer film 20 having the refractiveindex n₂ and the thickness Cd₂.

Each of the first layer film 14, the third layer film 16, the fifthlayer film 18 and the seventh layer film 20 is an alumina layer, andeach of the second layer film 15, the fourth layer film 17 and the sixthlayer film 19 is a tantalum oxide layer. Thus, the coating film 21according to the third preferred embodiment is composed of two materiallayers: the alumina layer and the tantalum oxide layer.

In the third preferred embodiment, the second and third layer films 15and 16, the fourth and fifth layer films 17 and 18, and the sixth andseventh layer films 19 and 20 constitute unit layer pairs each composedof the alumina layer and the tantalum oxide layer arranged in a stackedrelation. The reference characters A, B and C which determine thethicknesses of the second to seventh layer films 15 to 20 of the coatingfilm 21 designate parameters individually determined for the respectiveunit layer pairs and each indicating a contribution ratio of thethickness of a corresponding unit layer pair to the thickness of theentire coating film 21. The reference character O which determines thethickness of the first layer film 14 designates a parameter indicating acontribution ratio of the thickness of the first layer film 14 to thethickness of the entire coating film 21. The reference characters d₁ andd₂ according to the third preferred embodiment also designate basicthicknesses individually determined for the respective material layers.

The amounts of phase change for the first to seventh layer films 14 to20 are designated by Oφ₂, Aφ₁, Aφ₂, Bφ₁, Bφ₂, Cφ₁ and Cφ₂, respectively,using Equations (5a) and (5b). Therefore, the elements m₁₁, m₁₂, m₂₁ andm₂₂ of a characteristic matrix for the coating film 21 according to thethird preferred embodiment satisfy the following determinant:$\begin{matrix}\begin{matrix}{\begin{bmatrix}m_{11} & m_{12} \\m_{21} & m_{22}\end{bmatrix} = {\begin{bmatrix}{\cos\quad O\quad\phi_{2}} & {{- \frac{\mathbb{i}}{n_{2}}}\sin\quad O\quad\phi_{2}} \\{{- {\mathbb{i}}}\quad n_{2}\sin\quad O\quad\phi_{2}} & {\cos\quad O\quad\phi_{2}}\end{bmatrix} \times}} \\{{\begin{bmatrix}{\cos\quad A\quad\phi_{1}} & {{- \frac{\mathbb{i}}{n_{1}}}\sin\quad A\quad\phi_{1}} \\{{- {\mathbb{i}}}\quad n_{1}\sin\quad A\quad\phi_{1}} & {\cos\quad A\quad\phi_{1}}\end{bmatrix}\begin{bmatrix}{\cos\quad A\quad\phi_{2}} & {{- \frac{\mathbb{i}}{n_{2}}}\sin\quad A\quad\phi_{2}} \\{{- {\mathbb{i}n}_{2}}\sin\quad A\quad\phi_{2}} & {\cos\quad A\quad\phi_{2}}\end{bmatrix}} \times} \\{{\begin{bmatrix}{\cos\quad B\quad\phi_{1}} & {{- \frac{\mathbb{i}}{n_{1}}}\sin\quad B\quad\phi_{1}} \\{{- {\mathbb{i}}}\quad n_{1}\sin\quad B\quad\phi_{1}} & {\cos\quad B\quad\phi_{1}}\end{bmatrix}\begin{bmatrix}{\cos\quad B\quad\phi_{2}} & {{- \frac{\mathbb{i}}{n_{2}}}\sin\quad B\quad\phi_{2}} \\{{- {\mathbb{i}}}\quad n_{2}\sin\quad B\quad\phi_{2}} & {\cos\quad B\quad\phi_{2}}\end{bmatrix}} \times} \\{\begin{bmatrix}{\cos\quad C\quad\phi_{1}} & {{- \frac{\mathbb{i}}{n_{1}}}\sin\quad C\quad\phi_{1}} \\{{- {\mathbb{i}}}\quad n_{1}\sin\quad C\quad\phi_{1}} & {\cos\quad C\quad\phi_{1}}\end{bmatrix}\begin{bmatrix}{\cos\quad C\quad\phi_{2}} & {{- \frac{\mathbb{i}}{n_{2`}}}\sin\quad C\quad\phi_{2}} \\{{- {\mathbb{i}}}\quad n_{2}\sin\quad C\quad\phi_{2}} & {\cos\quad C\quad\phi_{2}}\end{bmatrix}}\end{matrix} & \left( {7c} \right)\end{matrix}$

For designing the thickness of the coating film 21 so that the value ofthe amplitude reflectivity r is an imaginary number, the third preferredembodiment previously determines the values of the parameters A, B, Cand O, and executes Steps s1 to s5 using Equations (6) and (7c)described above to determine the values of the basic amounts of phasechange φ₁ and φ₂. Thereafter, Step s6 is executed to determine thevalues of the basic thicknesses d₁ and d₂, and the thicknesses of therespective layers of the coating film 21 are determined using thepreviously determined values of the parameters A, B, C and O and thevalues of the basic thicknesses d₁ and d₂ determined in Step s6. If thedesigned characteristic of the coating film 21 is insufficient, thebasic thicknesses d₁ and d₂ are determined again by changing theparameters A, B, C and O, and the thickness of the coating film 21 isdesigned again.

In the third preferred embodiment, the effective refractive index n_(c)of the semiconductor photonic element 1 is “3.37.” The refractiveindices n₁ and n₂ are “2.057” and “1.620,” respectively. The designwavelength λ_(t) is 1310 nm.

For designing the thickness of the coating film 21 having the powerreflectivity R of 4% (R_(t)=4%) when the wavelength λ equals the designwavelength of 1310 nm under the above-mentioned conditions, a pointwhich provides a phase angle θ of 225 degrees is selected, for example,in Step s1 so that the reflection amplitude vector rv is located in thethird quadrant (or the lower left quadrant) of the complex plane. Then,the values of the real and imaginary parts r_(r) and r_(i) of thecomplex number inputted as the value of the amplitude reflectivity r are“−0.141421356” and “−0.141421356,” respectively, in Step s2.

When O=0.10, A=1.80, B=2.00 and C=2.00 are set, the basic amounts ofphase change φ₁ and φ₂ determined in Step s5 are “0.992102” and“0.536659,” respectively. Accordingly, the thicknesses of the first toseventh layer films 14 to 20 determined in Step s6 are 6.91 nm, 181.00nm, 124.32 nm, 201.12 nm, 138.14 nm, 201.12 nm and 138.14 nm,respectively.

FIG. 16 shows the wavelength dependence of the power reflectivity R ofthe coating film 21 thus designed. As illustrated in FIG. 16, the powerreflectivity R is 4% when the wavelength λ equals the design wavelengthof 1310 nm. A wavelength band for which the power reflectivity R isapproximately equal to the design reflectivity of 4% is wide. The powerreflectivity R falls within a range from 4.0% to 6.0% for the wavelengthλ ranging from 1116 nm to 1383 nm.

When the allowable reflectivity range is, for example, ±2% from thedesign reflectivity of 4%, the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range is alsofrom 1116 nm to 1383 nm, to provide a wavelength bandwidth W of 267 nm.The center wavelength λ_(c) of the wavelength band is 1250 nm. The valueobtained by dividing the wavelength bandwidth W by the center wavelengthλ_(c) is approximately 0.216, which is greater than 0.06. It may be saidthat the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range is a wide band.

Because the thicknesses of the first to seventh layer films 14 to 20take on the above-mentioned values, the optical thickness t of thecoating film 21, i.e. the sum of the products of the refractive indicesand thicknesses of the respective layers of the coating film 21, is1859.89 nm. This value is approximately 5.94 times the value t_(r) (313nm) which is a quarter of the center wavelength λ_(c). Thus, the coatingfilm 21 is a very thick film. This improves heat dissipationcharacteristics at the end surface 1 b of the semiconductor photonicelement 1 to suppress the increase in temperature of the end surface 1b.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 4% for the wavelength λ equal to λ_(c) (1250 nm)when the coating film 2 having the refractive index n_(f) of “1.4989” asin the first preferred embodiment and a thickness d_(f) of five times208.49 nm, i.e. five times λ_(c)/(4n_(f)), is provided on the endsurface 1 b of the semiconductor photonic element 1. FIG. 17 shows thewavelength dependence of the power reflectivity R in this case. Asillustrated in FIG. 17, the wavelength band for which the powerreflectivity R falls within ±2% from the design reflectivity of 4% isfrom 1213 nm to 1290 nm, to provide a wavelength bandwidth W_(r) of 77nm.

Thus, the wavelength bandwidth W (267 nm) for the coating film 21 of thethird preferred embodiment is greater than the wavelength bandwidthW_(r) (77 nm) for the coating film 2 shown in FIG. 2.

Because the wavelength λ of light propagating through the active layer 1a is sometimes varied, it is desirable that the center wavelength λ_(c)of the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range be equal to or close to the designwavelength λ_(t). For example, the center wavelength λ_(c) takes on avalue close to the design wavelength λ_(t) of 1310 nm when the thicknessof the coating film 21 is determined using the basic thicknesses d₁ andd₂ obtained by setting the basic amounts of phase change φ₁ and φ₂ at“0.992102” and “0.536659,” respectively, in a similar manner to theabove instance and substituting 1374 nm, rather than the designwavelength λ_(t), for λ in Equations (5a) and (5b). FIG. 18 shows thewavelength dependence of the power reflectivity R in this case.

As illustrated in FIG. 18, the power reflectivity R falls within a rangefrom 4.0% to 6.0% for the wavelength λ ranging from 1170 nm to 1451 nm.The wavelength band for which the power reflectivity R falls within ±2%from the design reflectivity of 4% is also from 1170 nm to 1451 nm, toprovide a wavelength bandwidth W of 281 nm. The center wavelength λ_(c)of the wavelength band is 1311 nm, which is very close to the designwavelength of 1310 nm. The value obtained by dividing the wavelengthbandwidth W (281 nm) by the center wavelength λ_(c) (1311 nm) isapproximately 0.214, which is greater than 0.06 and shows that thewavelength band for which the power reflectivity R falls within theallowable reflectivity range is a wide band.

In this case, the thicknesses of the first to seventh layer films 14 to20 of the coating film 21 are 7.24 nm, 189.85 nm, 130.40 nm, 210.94 nm,144.88 nm, 210.94 nm and 144.88 nm, respectively. The optical thicknesst of the coating film 21 is 1950.72 nm. This value is approximately 5.95times the value t_(r) (328 nm) which is a quarter of the centerwavelength λ_(c). Thus, the coating film 21 is a very thick film.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 4% for the wavelength λ equal to λ_(c) (1311 nm)when the coating film 2 having the above-mentioned refractive indexn_(f) of “1.4989” and a thickness d_(f) of five times 218.66 nm, i.e.five times λ_(c)/(4n_(f)), is provided on the end surface 1 b of thesemiconductor photonic element 1. FIG. 19 shows the wavelengthdependence of the power reflectivity R in this case. As illustrated inFIG. 19, the wavelength band for which the power reflectivity R fallswithin ±2% from the design reflectivity of 4% is from 1271 nm to 1352nm, to provide a wavelength bandwidth W_(r) of 81 nm.

Thus, even when the center wavelength λ_(c) is made closer to the designwavelength λ_(t), the wavelength bandwidth W (281 nm) for the coatingfilm 21 is greater than the wavelength bandwidth W_(r) (81 nm) for thecoating film 2 shown in FIG. 2.

Although the center wavelength λ_(c) is close to the design wavelengthλ_(t) in the above instance, the center wavelength λ_(c) may be madeexactly equal to the design wavelength λ_(t) by adjusting the values ofthe parameters A, B, C and O or the value substituted for λ in Equations(5a) and (5b), as in the first and second preferred embodiments.

Fourth Preferred Embodiment

FIG. 20 is a side view showing a structure of the semiconductor photonicdevice according to the fourth preferred embodiment of the presentinvention. A coating film 29 having a seven-layer structure is providedon the end surface 1 b of the semiconductor photonic element 1 in thefourth preferred embodiment.

As illustrated in FIG. 20, the coating film 29 according to the fourthpreferred embodiment of the present invention includes: a first layerfilm 22 having a refractive index n₃ and a thickness d₃; a second layerfilm 23 having the refractive index n₁ and the thickness Ad₁; a thirdlayer film 24 having the refractive index n₂ and the thickness Ad₂; afourth layer film 25 having the refractive index n₁ and the thicknessBd₁; a fifth layer film 26 having the refractive index n₂ and thethickness Bd₂; a sixth layer film 27 having the refractive index n₁ andthe thickness Cd₁; and a seventh layer film 28 having the refractiveindex n₂ and the thickness Cd₂.

The first layer film 22 is an aluminum nitride layer. Each of the secondlayer film 23, the fourth layer film 25 and the sixth layer film 27 is atantalum oxide layer. Each of the third layer film 24, the fifth layerfilm 26 and the seventh layer film 28 is an alumina layer. Thus, thecoating film 29 according to the fourth preferred embodiment is composedof three material layers: the aluminum nitride layer, the tantalum oxidelayer, and the alumina layer.

In the fourth preferred embodiment, the second and third layer films 23and 24, the fourth and fifth layer films 25 and 26, and the sixth andseventh layer films 27 and 28 constitute unit layer pairs each composedof the tantalum oxide layer and the alumina layer arranged in a stackedrelation. The reference characters A, B and C which determine thethicknesses of the second to seventh layer films 23 to 28 of the coatingfilm 29 designate parameters individually determined for the respectiveunit layer pairs and each indicating a contribution ratio of thethickness of a corresponding unit layer pair to the thickness of theentire coating film 29. The reference characters d₁ and d₂ according tothe fourth preferred embodiment also designate basic thicknessesindividually determined for the respective material layers.

The amounts of phase change for the second to seventh layer films 23 to28 are designated by Aφ₁, Aφ₂, Bφ₁, Bφ₂, Cφ₁ and Cφ₂, respectively,using Equations (5a) and (5b). The amount of phase change φ₃ for thefirst layer film 22 is expressed by: $\begin{matrix}{\phi_{3} = {\frac{2\pi}{\lambda}n_{3}d_{3}}} & (9)\end{matrix}$

Therefore, the elements m₁₁, m₁₂, m₂₁ and m₂₂ of a characteristic matrixfor the coating film 29 according to the fourth preferred embodimentsatisfy the following determinant: $\begin{matrix}\begin{matrix}{\begin{bmatrix}m_{11} & m_{12} \\m_{21} & m_{22}\end{bmatrix} = {\begin{bmatrix}{\cos\quad\phi_{3}} & {{- \frac{\mathbb{i}}{n_{3}}}\sin\quad\phi_{3}} \\{{- {\mathbb{i}}}\quad n_{3}\sin\quad\phi_{3}} & {\cos\quad\phi_{3}}\end{bmatrix} \times}} \\{{\begin{bmatrix}{\cos\quad A\quad\phi_{1}} & {{- \frac{\mathbb{i}}{n_{1}}}\sin\quad A\quad\phi_{1}} \\{{- {\mathbb{i}}}\quad n_{1}\sin\quad A\quad\phi_{1}} & {\cos\quad A\quad\phi_{1}}\end{bmatrix}\begin{bmatrix}{\cos\quad A\quad\phi_{2}} & {{- \frac{\mathbb{i}}{n_{2}}}\sin\quad A\quad\phi_{2}} \\{{- {\mathbb{i}n}_{2}}\sin\quad A\quad\phi_{2}} & {\cos\quad A\quad\phi_{2}}\end{bmatrix}} \times} \\{{\begin{bmatrix}{\cos\quad B\quad\phi_{1}} & {{- \frac{\mathbb{i}}{n_{1}}}\sin\quad B\quad\phi_{1}} \\{{- {\mathbb{i}}}\quad n_{1}\sin\quad B\quad\phi_{1}} & {\cos\quad B\quad\phi_{1}}\end{bmatrix}\begin{bmatrix}{\cos\quad B\quad\phi_{2}} & {{- \frac{\mathbb{i}}{n_{2}}}\sin\quad B\quad\phi_{2}} \\{{- {\mathbb{i}}}\quad n_{2}\sin\quad B\quad\phi_{2}} & {\cos\quad B\quad\phi_{2}}\end{bmatrix}} \times} \\{\begin{bmatrix}{\cos\quad C\quad\phi_{1}} & {{- \frac{\mathbb{i}}{n_{1}}}\sin\quad C\quad\phi_{1}} \\{{- {\mathbb{i}}}\quad n_{1}\sin\quad C\quad\phi_{1}} & {\cos\quad C\quad\phi_{1}}\end{bmatrix}\begin{bmatrix}{\cos\quad C\quad\phi_{2}} & {{- \frac{\mathbb{i}}{n_{2`}}}\sin\quad C\quad\phi_{2}} \\{{- {\mathbb{i}}}\quad n_{2}\sin\quad C\quad\phi_{2}} & {\cos\quad C\quad\phi_{2}}\end{bmatrix}}\end{matrix} & \left( {7d} \right)\end{matrix}$

For designing the thickness of the coating film 29 so that the value ofthe amplitude reflectivity r is an imaginary number, the fourthpreferred embodiment previously determines the values of the parametersA, B and C, and previously determines the thickness d₃ of the firstlayer film 22, thereby to handle the value of the amount of phase changeφ₃ as a known value. Steps s1 to s5 are executed using Equations (6) and(7d) described above to determine the values of the basic amounts ofphase change φ₁ and φ₂. Thereafter, Step s6 is executed to determine thevalues of the basic thicknesses d₁ and d₂, and the thicknesses of thesecond to seventh layer films 23 to 28 of the coating film 29 aredetermined using the previously determined values of the parameters A, Band C and the values of the basic thicknesses d₁ and d₂ determined inStep s6. If the designed characteristic of the coating film 29 isinsufficient, the basic thicknesses d₁ and d₂ are determined again bychanging the parameters A, B and C or the thickness d₃, and thethickness of the coating film 29 is designed again.

The method of determining the thickness when the coating film iscomposed of the three material layers is described above. However, ifthe coating film is composed of four or more material layers, thismethod is capable of determining the thicknesses of the respectivelayers of the coating film in a similar manner to the fourth preferredembodiment by handling the thicknesses of some of the plurality oflayers of the coating film which are included in the first and secondmaterial layers as unknown values while handling the thicknesses of theremaining layers which are included in the third and its subsequentmaterial layers as known values.

In the fourth preferred embodiment, the effective refractive index n_(c)of the semiconductor photonic element 1 is “3.37.” The refractiveindices n₁ to n₃ are “2.057,” “1.620” and “2.072,” respectively. Thedesign wavelength λ_(t) is 1550 nm.

For designing the thicknesses of the respective layers of the coatingfilm 29 having the power reflectivity R of 4% when the wavelength λequals the design wavelength of 1550 nm under the above-mentionedconditions, a point which provides a phase angle θ of 330 degrees isselected, for example, in Step s1 so that the reflection amplitudevector rv is located in the fourth quadrant (or the lower rightquadrant) of the complex plane. Then, the values of the real andimaginary parts r_(r) and r_(i) of the complex number inputted as thevalue of the amplitude reflectivity r are “+0.17320508” and “−0.1,”respectively, in Step s2.

When A=1.69, B=1.65, C=2.08 and d₃=7.5 nm are set, the basic amounts ofphase change φ₁ and φ₂ determined in Step s5 are “1.33612” and“0.478116,” respectively. Accordingly, the thicknesses of the second toseventh layer films 23 to 28 determined in Step s6 are 270.80 nm, 123.04nm, 264.39 nm, 120.13 nm, 333.29 nm and 151.44 nm, respectively.

FIG. 21 shows the wavelength dependence of the power reflectivity R ofthe coating film 29 thus designed. As illustrated in FIG. 21, the powerreflectivity R is 4% when the wavelength λ equals the design wavelengthof 1550 nm. A wavelength band for which the power reflectivity R isapproximately equal to the design reflectivity of 4% is wide. The powerreflectivity R falls within a range from 2.1% to 6.0% for the wavelengthλ ranging from 1420 nm to 1898 nm.

When the allowable reflectivity range is, for example, ±2% from thedesign reflectivity of 4%, the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range is alsofrom 1420 nm to 1898 nm, to provide a wavelength bandwidth W of 478 nm.The center wavelength λ_(c) of the wavelength band is 1659 nm. The valueobtained by dividing the wavelength bandwidth W by the center wavelengthλ_(c) is approximately 0.288, which is greater than 0.06. It may be saidthat the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range is a wide band.

Because the thicknesses of the first to seventh layer films 22 to 28take on the above-mentioned values, the optical thickness t of thecoating film 29, i.e. the sum of the products of the refractive indicesand thicknesses of the respective layers of the coating film 29, is2441.27 nm. This value is approximately 5.88 times the value t_(r) (415nm) which is a quarter of the center wavelength λ_(c). Thus, the coatingfilm 29 is a very thick film. This improves heat dissipationcharacteristics at the end surface 1 b of the semiconductor photonicelement 1 to suppress the increase in temperature of the end surface 1b.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 4% for the wavelength λ equal to λ_(c) (1659 nm)when the coating film 2 having the refractive index n_(f) of “1.4989” asin the first preferred embodiment and a thickness d_(f) of five times276.70 nm, i.e. five times λ_(c)/(4n_(f)), is provided on the endsurface 1 b of the semiconductor photonic element 1. FIG. 22 shows thewavelength dependence of the power reflectivity R in this case. Asillustrated in FIG. 22, the wavelength band for which the powerreflectivity R falls within ±2% from the design reflectivity of 4% isfrom 1609 nm to 1712 nm, to provide a wavelength bandwidth W_(r) of 103nm.

Thus, the wavelength bandwidth W (478 nm) for the coating film 29 of thefourth preferred embodiment is greater than the wavelength bandwidthW_(r) (103 nm) for the coating film 2 shown in FIG. 2.

Because the wavelength λ of light propagating through the active layer 1a is sometimes varied, it is desirable that the center wavelength λ_(c)of the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range be equal to or close to the designwavelength λ_(t). For example, when d₃=7.02 nm in the above instance,the basic amounts of phase change φ₁ and φ₂ are “1.33612” and“0.478115,” respectively. The center wavelength λ_(c) takes on a valueequal to the design wavelength of 1550 nm when the thickness of thecoating film 29 is determined using the basic thicknesses d₁ and d₂obtained by substituting the above-mentioned values of the basic amountsof phase change φ₁ and φ₂ into Equations (5a) and (5b) and substituting1451 nm, rather than the design wavelength λ_(t), for λ in Equations(5a) and (5b). FIG. 23 shows the wavelength dependence of the powerreflectivity R in this case.

As illustrated in FIG. 23, the power reflectivity R falls within a rangefrom 2.0% to 6.0% for the wavelength λ ranging from 1322 nm to 1777 nm.The wavelength band for which the power reflectivity R falls within ±2%from the design reflectivity of 4% is also from 1322 nm to 1777 nm, toprovide a wavelength bandwidth W of 455 nm. The center wavelength λ_(c)of the wavelength band is 1550 nm, which is equal to the designwavelength of 1550 nm. The value obtained by dividing the wavelengthbandwidth W (455 nm) by the center wavelength λ_(c) (1550 nm) isapproximately 0.294, which is greater than 0.06 and shows that thewavelength band for which the power reflectivity R falls within theallowable reflectivity range is a wide band.

In this case, the thicknesses of the first to seventh layer films 22 to28 of the coating film 29 are 7.02 nm, 253.50 nm, 115.18 nm, 247.50 nm,112.46 nm, 312.01 nm and 141.77 nm, respectively. The optical thicknesst of the coating film 29 is 2285.35 nm. This value is approximately 5.89times the value t_(r) (388 nm) which is a quarter of the centerwavelength λ_(c). Thus, the coating film 29 is a very thick film.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 4% for the wavelength λ equal to λ_(c) (1550 nm)when the coating film 2 having the above-mentioned refractive indexn_(f) of “1.4989” and a thickness d_(f) of five times 258.52 nm, i.e.five times λ_(c)/(4n_(f)), is provided on the end surface 1 b of thesemiconductor photonic element 1. FIG. 24 shows the wavelengthdependence of the power reflectivity R in this case. As illustrated inFIG. 24, the wavelength band for which the power reflectivity R fallswithin ±2% from the design reflectivity of 4% is from 1503 nm to 1600nm, to provide a wavelength bandwidth W_(r) of 97 nm.

Thus, even when the center wavelength λ_(c) is made equal to the designwavelength λ_(t), the wavelength bandwidth W (455 nm) for the coatingfilm 29 is greater than the wavelength bandwidth W_(r) (97 nm) for thecoating film 2 shown in FIG. 2.

Fifth Preferred Embodiment

FIG. 25 is a side view showing a structure of the semiconductor photonicdevice according to the fifth preferred embodiment of the presentinvention. A coating film 38 having an eight-layer structure is providedon the end surface 1 b of the semiconductor photonic element 1 in thefifth preferred embodiment.

As illustrated in FIG. 25, the coating film 38 according to the fifthpreferred embodiment of the present invention includes: a first layerfilm 30 having the refractive index n₁ and the thickness Ad₁; a secondlayer film 31 having the refractive index n₂ and the thickness Ad₂; athird layer film 32 having the refractive index n₁ and the thicknessBd₁; a fourth layer film 33 having the refractive index n₂ and thethickness Bd₂; a fifth layer film 34 having the refractive index n₁ andthe thickness Cd₁; a sixth layer film 35 having the refractive index n₂and the thickness Cd₂; a seventh layer film 36 having the refractiveindex n₁ and a thickness Dd₁; and an eighth layer film 37 having therefractive index n₂ and a thickness Dd₂.

Each of the first layer film 30, the third layer film 32, the fifthlayer film 34 and the seventh layer film 36 is a tantalum oxide layer,and each of the second layer film 31, the fourth layer film 33, thesixth layer film 35 and the eighth layer film 37 is a silicon oxidelayer. Thus, the coating film 38 according to the fifth preferredembodiment is composed of two material layers: the tantalum oxide layerand the silicon oxide layer.

In the fifth preferred embodiment, the first and second layer films 30and 31, the third and fourth layer films 32 and 33, the fifth and sixthlayer films 34 and 35, and the seventh and eighth layer films 36 and 37constitute unit layer pairs each composed of the tantalum oxide layerand the silicon oxide layer arranged in a stacked relation. In otherwords, the coating film 38 includes four unit layer pairs each composedof the tantalum oxide layer and the silicon oxide layer arranged in astacked relation. The reference characters A, B, C and D which determinethe thicknesses of the respective layers of the coating film 38designate parameters individually determined for the respective unitlayer pairs and each indicating a contribution ratio of the thickness ofa corresponding unit layer pair to the thickness of the entire coatingfilm 38. The reference characters d₁ and d₂ according to the fifthpreferred embodiment also designate basic thicknesses individuallydetermined for the respective material layers in a similar manner to thefirst preferred embodiment.

The amounts of phase change for the first to eighth layer films 30 to 37are designated by Aφ₁, Aφ₂, Bφ₁, Bφ₂, Cφ₁, Cφ₂, Dφ₁ and Dφ₂,respectively, using Equations (5a) and (5b). Therefore, the elementsm₁₁, m₁₂, m₂₁ and m₂₂ of a characteristic matrix for the coating film 38according to the fifth preferred embodiment satisfy the followingdeterminant: $\begin{matrix}{\begin{bmatrix}m_{11} & m_{12} \\m_{21} & m_{22}\end{bmatrix}{\quad{= {\begin{bmatrix}{\cos\quad A\quad\phi_{1}} & {{- \frac{i}{n_{1}}}\sin\quad A\quad\phi_{1}} \\{{- {in}_{1}}\sin\quad A\quad\phi_{1}} & {\cos\quad A\quad\phi_{1}}\end{bmatrix}{\quad{\begin{bmatrix}{\cos\quad A\quad\phi_{2}} & {{- \frac{i}{n_{2}}}\sin\quad A\quad\phi_{2}} \\{{- {in}_{2}}\sin\quad A\quad\phi_{2}} & {\cos\quad A\quad\phi_{2}}\end{bmatrix} \times {\quad{\begin{bmatrix}{\cos\quad B\quad\phi_{1`}} & {{- \frac{i}{n_{1}}}\sin\quad B\quad\phi_{1}} \\{{- {in}_{1}}\sin\quad B\quad\phi_{1}} & {\cos\quad B\quad\phi_{1}}\end{bmatrix}{\quad{\begin{bmatrix}{\cos\quad B\quad\phi_{2}} & {{- \frac{i}{n_{2}}}\sin\quad B\quad\phi_{2}} \\{{- {in}_{2}}\sin\quad B\quad\phi_{2}} & {\cos\quad B\quad\phi_{2}}\end{bmatrix} \times {\quad{\begin{bmatrix}{\cos\quad C\quad\phi_{1}} & {{- \frac{i}{n_{1}}}\sin\quad C\quad\phi_{1}} \\{{- {in}_{1}}\sin\quad C\quad\phi_{1}} & {\cos\quad C\quad\phi_{1}}\end{bmatrix}{\quad{\begin{bmatrix}{\cos\quad C\quad\phi_{2}} & {{- \frac{i}{n_{2}}}\sin\quad C\quad\phi_{2}} \\{{- {in}_{2}}\sin\quad C\quad\phi_{2}} & {\cos\quad C\quad\phi_{2}}\end{bmatrix} \times {\quad{\begin{bmatrix}{\cos\quad D\quad\phi_{1}} & {{- \frac{i}{n_{1}}}\sin\quad D\quad\phi_{1}} \\{{- {in}_{1}}\sin\quad D\quad\phi_{1}} & {\cos\quad D\quad\phi_{1}}\end{bmatrix}{\quad\begin{bmatrix}{\cos\quad D\quad\phi_{2}} & {{- \frac{i}{n_{2}}}\sin\quad D\quad\phi_{2}} \\{{- {in}_{2}}\sin\quad D\quad\phi_{2}} & {\cos\quad D\quad\phi_{2}}\end{bmatrix}}}}}}}}}}}}}}}}}} & \left( {7e} \right)\end{matrix}$

For designing the thickness of the coating film 38 so that the value ofthe amplitude reflectivity r is an imaginary number the fifth preferredembodiment previously determines the values of the parameters A to D,and executes Steps s1 to s5 using Equations (6) and (7d) described aboveto determine the values of the basic amounts of phase change φ₁ and φ₂.Thereafter, Step s6 is executed to determine the values of the basicthicknesses d₁ and d₂, and the thicknesses of the respective layers ofthe coating film 38 are determined using the previously determinedvalues of the parameters A to D and the values of the basic thicknessesd₁ and d₂ determined in Step s6. If the designed characteristic of thecoating film 38 is insufficient, the basic thicknesses d₁ and d₂ aredetermined again by changing the parameters A to D, and the thickness ofthe coating film 38 is designed again.

Because GaN (gallium nitride) based semiconductor is employed in thesemiconductor photonic element 1 of the fifth preferred embodiment, theeffective refractive index n_(c) of the semiconductor photonic element 1is “2.50.” The design wavelength λ₁ is 410 nm. The refractive index n₁of the tantalum oxide layer is greater than n_(c) ^(1/2) (=1.581), andis set at “2.128” in consideration for wavelength dispersion because thedesign wavelength λ_(t) is as short as 410 nm. The refractive index n₂of the silicon oxide layer is “1.480” which is less than n_(c) ^(1/2).

For designing the thickness of the coating film 38 having the powerreflectivity R of 4% when the wavelength λ equals the design wavelengthof 410 nm under the above-mentioned conditions, a point which provides aphase angle θ of 45 degrees is selected, for example, in Step s1 so thatthe reflection amplitude vector rv is located in the first quadrant (orthe upper right quadrant) of the complex plane. Then, the values of thereal and imaginary parts r_(r) and r_(i) of the complex number inputtedas the value of the amplitude reflectivity r are “+0.141421356” and“+0.141421356,” respectively, in Step s2.

When A=1.38, B=2.30, C=2.00 and D=2.00 are set, the basic amounts ofphase change φ₁ and φ₂ determined in Step s5 are “1.56840” and“0.526521,” respectively. Accordingly, the thicknesses of the first toeighth layer films 30 to 37 determined in Step s6 are 66.37 nm, 32.04nm, 110.62 nm, 53.39 nm, 96.19 nm, 46.43 nm, 96.19 nm and 46.43 nm,respectively.

FIG. 26 shows the wavelength dependence of the power reflectivity R ofthe coating film 38 thus designed. As illustrated in FIG. 26, the powerreflectivity R is 4% when the wavelength λ equals the design wavelengthof 410 nm. A wavelength band for which the power reflectivity R isapproximately equal to the design reflectivity of 4% is wide. The powerreflectivity R falls within a range from 3.9% to 6.0% for the wavelengthλ ranging from 386 nm to 488 nm.

When the allowable reflectivity range is, for example, ±2% from thedesign reflectivity of 4%, the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range is alsofrom 386 nm to 488 nm, to provide a wavelength bandwidth W of 102 nm.The center wavelength λ_(c) of the wavelength band is 437 nm. The valueobtained by dividing the wavelength bandwidth W by the center wavelengthλ_(c) is approximately 0.233, which is greater than 0.06. It may be saidthat the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range is a wide band.

Because the thicknesses of the first to eighth layer films 30 to 37 takeon the above-mentioned values, the optical thickness t of the coatingfilm 38, i.e. the sum of the products of the refractive indices andthicknesses of the respective layers of the coating film 38, is 1049.89nm. This value is approximately 9.63 times the value t_(r) (109 nm)which is a quarter of the center wavelength λ_(c). Thus, the coatingfilm 38 is a very thick film. This improves heat dissipationcharacteristics at the end surface 1 b of the semiconductor photonicelement 1 to suppress the increase in temperature of the end surface 1b.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R takes on a relative minimum value of 4% for thewavelength λ equal to λ_(c) (437 nm) when the coating film 2 having arefractive index n_(f) of “1.291” and a thickness d_(f) of five times84.62 nm, i.e. five times λ_(c)/(4n_(f)), is provided on the end surface1 b of the semiconductor photonic element 1. FIG. 27 shows thewavelength dependence of the power reflectivity R in this case. Asillustrated in FIG. 27, the wavelength band for which the powerreflectivity R falls within ±2% from the design reflectivity of 4% isfrom 419 nm to 457 nm, to provide a wavelength bandwidth W_(r) of 38 nm.

The above-mentioned value “1.291” of the refractive index n_(f) isobtained by substituting R_(t)=0.04 and n_(c)=2.50 into Equation (8).

Thus, the wavelength bandwidth W (102 nm) for the coating film 38 of thefifth preferred embodiment is greater than the wavelength bandwidthW_(r) (38 nm) for the coating film 2 shown in FIG. 2.

Because the wavelength λ of light propagating through the active layer 1a is sometimes varied, it is desirable that the center wavelength λ_(c)of the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range be equal to or close to the designwavelength λ_(t). For example, the center wavelength λ_(c) takes on avalue equal to the design wavelength λ_(t) of 410 nm when the thicknessof the coating film 38 is determined using the basic thicknesses d₁ andd₂ obtained by setting the basic amounts of phase change φ₁ and φ₂ at“1.56840” and “0.526521,” respectively, in a similar manner to the aboveinstance and substituting 384 nm, rather than the design wavelengthλ_(t), for λ in Equations (5a) and (5b). FIG. 28 shows the wavelengthdependence of the power reflectivity R in this case.

As illustrated in FIG. 28, the power reflectivity R falls within a rangefrom 3.9% to 6.0% for the wavelength λ ranging from 362 nm to 457 nm.The wavelength band for which the power reflectivity R falls within ±2%from the design reflectivity of 4% is also from 362 nm to 457 nm, toprovide a wavelength bandwidth W of 95 nm. The center wavelength λ_(c)of the wavelength band is 410 nm, which is equal to the designwavelength of 410 nm. The value obtained by dividing the wavelengthbandwidth W (95 nm) by the center wavelength λ_(c) (410 nm) isapproximately 0.224, which is greater than 0.06 and shows that thewavelength band for which the power reflectivity R falls within theallowable reflectivity range is a wide band.

In this case, the thicknesses of the first to eighth layer films 30 to37 of the coating film 38 are 62.16 nm, 30.00 nm, 103.60 nm, 50.00 nm,90.09 nm, 43.48 nm, 90.09 nm and 43.48 nm, respectively. The opticalthickness t of the coating film 38 is 983.26 nm. This value isapproximately 9.55 times the value t_(r) (103 nm) which is a quarter ofthe center wavelength λ_(c). Thus, the coating film 38 is a very thickfilm.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 4% for the wavelength λ equal to λ_(c) (410 nm)when the coating film 2 having the above-mentioned refractive indexn_(f) of “1.291” and a thickness d_(f) of five times 79.40 nm, i.e. fivetimes λ_(c)/(4n_(f)), is provided on the end surface 1 b of thesemiconductor photonic element 1. FIG. 29 shows the wavelengthdependence of the power reflectivity R in this case. As illustrated inFIG. 29, the wavelength band for which the power reflectivity R fallswithin ±2% from the design reflectivity of 4% is from 393 nm to 429 nm,to provide a wavelength bandwidth W_(r) of 36 nm.

Thus, even when the center wavelength λ_(c) is made equal to the designwavelength λ_(t), the wavelength bandwidth W (95 nm) for the coatingfilm 38 is greater than the wavelength bandwidth W_(r) (36 nm) for thecoating film 2 shown in FIG. 2.

Sixth Preferred Embodiment

FIG. 30 is a side view showing a structure of the semiconductor photonicdevice according to the sixth preferred embodiment of the presentinvention. A coating film 47 having an eight-layer structure is providedon the end surface 1 b of the semiconductor photonic element 1 in thesixth preferred embodiment.

As illustrated in FIG. 30, the coating film 47 according to the sixthpreferred embodiment of the present invention includes: a first layerfilm 39 having the refractive index n₃ and the thickness d₃; a secondlayer film 40 having the refractive index n₂ and the thickness Ad₂; athird layer film 41 having the refractive index n₁ and the thicknessBd₁; a fourth layer film 42 having the refractive index n₂ and thethickness Bd₂; a fifth layer film 43 having the refractive index n₁ andthe thickness Cd₁; a sixth layer film 44 having the refractive index n₂and the thickness Cd₂; a seventh layer film 45 having the refractiveindex n₁ and the thickness Dd₁; and an eighth layer film 46 having therefractive index n₂ and the thickness Dd₂.

The first layer film 39 is an aluminum nitride layer. Each of the secondlayer film 40, the fourth layer film 42, the sixth layer film 44 and theeighth layer film 46 is a silicon oxide layer. Each of the third layerfilm 41, the fifth layer film 43 and the seventh layer film 45 is atantalum oxide layer. Thus, the coating film 47 according to the sixthpreferred embodiment is composed of three material layers: the aluminumnitride layer, the silicon oxide layer, and the tantalum oxide layer.

In the sixth preferred embodiment, the third and fourth layer films 41and 42, the fifth and sixth layer films 43 and 44, and the seventh andeighth layer films 45 and 46 constitute unit layer pairs each composedof the tantalum oxide layer and the silicon oxide layer arranged in astacked relation. The reference characters B, C and D which determinethe thicknesses of the third to eighth layer films 41 to 46 of thecoating film 47 designate parameters individually determined for therespective unit layer pairs and each indicating a contribution ratio ofthe thickness of a corresponding unit layer pair to the thickness of theentire coating film 47. The reference character A which determines thethickness of the second layer film 40 designates a parameter indicatinga contribution ratio of the thickness of the second layer film 40 to thethickness of the entire coating film 47. The reference characters d₁ andd₂ according to the sixth preferred embodiment also designate basicthicknesses individually determined for the respective material layers.

The amounts of phase change for the second to eighth layer films 40 to46 are designated by Aφ₂, Bφ₁, Bφ₂, Cφ₁, Cφ₂, Dφ₁ and Dφ₂, respectively,using Equations (5a) and (5b). The amount of phase change φ₃ for thefirst layer film 39 is expressed by Equation (9) described above.Therefore, the elements m₁₁, m₁₂, m₂₁ and m₂₂ of a characteristic matrixfor the coating film 47 according to the sixth preferred embodimentsatisfy the following determinant: $\begin{matrix}{\begin{bmatrix}m_{11} & m_{12} \\m_{21} & m_{22}\end{bmatrix} = {\quad{\begin{bmatrix}{\cos\quad\phi_{3}} & {{- \frac{i}{n_{3}}}\sin\quad\phi_{3}} \\{{- {in}_{3}}\sin\quad\phi_{3}} & {\cos\quad\phi_{3}}\end{bmatrix}{\quad{\begin{bmatrix}{\cos\quad A\quad\phi_{2}} & {{- \frac{i}{n_{2}}}\sin\quad A\quad\phi_{2}} \\{{- {in}_{2}}\sin\quad A\quad\phi_{2}} & {\cos\quad A\quad\phi_{2}}\end{bmatrix} \times \begin{bmatrix}{\cos\quad B\quad\phi_{1}} & {{- \frac{i}{n_{1}}}\sin\quad B\quad\phi_{1}} \\{{- {in}_{1}}\sin\quad B\quad\phi_{1}} & {\cos\quad B\quad\phi_{1}}\end{bmatrix}{\quad{\begin{bmatrix}{\cos\quad B\quad\phi_{2}} & {{- \frac{i}{n_{2}}}\sin\quad B\quad\phi_{2}} \\{{- {in}_{2}}\sin\quad B\quad\phi_{2}} & {\cos\quad B\quad\phi_{2}}\end{bmatrix} \times \begin{bmatrix}{\cos\quad C\quad\phi_{1}} & {{- \frac{i}{n_{1}}}\sin\quad C\quad\phi_{1}} \\{{- {in}_{1}}\sin\quad C\quad\phi_{1}} & {\cos\quad C\quad\phi_{1}}\end{bmatrix}{\quad{\begin{bmatrix}{\cos\quad C\quad\phi_{2}} & {{- \frac{i}{n_{2}}}\sin\quad C\quad\phi_{2}} \\{{- {in}_{2}}\sin\quad C\quad\phi_{2}} & {\cos\quad C\quad\phi_{2}}\end{bmatrix} \times {\quad{\begin{bmatrix}{\cos\quad D\quad\phi_{1}} & {{- \frac{i}{n_{1}}}\sin\quad D\quad\phi_{1}} \\{{- {in}_{1}}\sin\quad D\quad\phi_{1}} & {\cos\quad D\quad\phi_{1}}\end{bmatrix}{\quad\begin{bmatrix}{\cos\quad D\quad\phi_{2}} & {{- \frac{i}{n_{2}}}\sin\quad D\quad\phi_{2}} \\{{- {in}_{2}}\sin\quad D\quad\phi_{2}} & {\cos\quad D\quad\phi_{2}}\end{bmatrix}}}}}}}}}}}}} & \left( {7f} \right)\end{matrix}$

For designing the thickness of the coating film 47 so that the value ofthe amplitude reflectivity r is an imaginary number, the sixth preferredembodiment previously determines the values of the parameters A to D,and previously determines the thickness d₃ of the first layer film 39,thereby to handle the value of the amount of phase change φ₃ as a knownvalue. Steps s1 to s5 are executed using Equations (6) and (7f)described above to determine the values of the basic amounts of phasechange φ₁ and φ₂. Thereafter, Step s6 is executed to determine thevalues of the basic thicknesses d₁ and d₂, and the thicknesses of thesecond to eighth layer films 40 to 46 of the coating film 47 aredetermined using the previously determined values of the parameters A toD and the values of the basic thicknesses d₁ and d₂ determined in Steps6. If the designed characteristic of the coating film 47 isinsufficient, the basic thicknesses d₁ and d₂ are determined again bychanging the parameters A to D or the thickness d₃, and the thickness ofthe coating film 47 is designed again.

In the sixth preferred embodiment, the effective refractive index n_(c)of the semiconductor photonic element 1 is “3.37.” The refractiveindices n₁ to n₃ are “2.057,” “1.480” and “2.072,” respectively. Thedesign wavelength λ_(t) is 650 nm.

For designing the thickness of the coating film 47 having the powerreflectivity R of 4% when the wavelength λ equals the design wavelengthof 650 nm under the above-mentioned conditions, a point which provides aphase angle θ of 135 degrees is selected, for example, in Step s1 sothat the reflection amplitude vector rv is located in the secondquadrant (or the upper left quadrant) of the complex plane. Then, thevalues of the real and imaginary parts r_(r) and r_(i) of the complexnumber inputted as the value of the amplitude reflectivity r are“−0.141421356” and “+0.141421356,” respectively, in Step s2.

When A=2.50, B=1.90, C=1.00, D=2.05 and d₃=40.0 nm are set, the basicamounts of phase change φ₁ and φ₂ determined in Step s5 are “0.674374”and “1.15311,” respectively. Accordingly, the thicknesses of the secondto eighth layer films 40 to 46 determined in Step s6 are 201.50 nm,64.44 nm, 153.14 nm, 33.92 nm, 80.60 nm, 69.53 nm and 165.23 nm,respectively.

FIG. 31 shows the wavelength dependence of the power reflectivity R ofthe coating film 47 thus designed. As illustrated in FIG. 31, the powerreflectivity R is 4% when the wavelength λ equals the design wavelengthof 650 nm. A wavelength band for which the power reflectivity R isapproximately equal to the design reflectivity of 4% is wide. The powerreflectivity R falls within a range from 4.0% to 6.0% for the wavelengthλ ranging from 630 nm to 736 nm.

When the allowable reflectivity range is, for example, ±2% from thedesign reflectivity of 4%, the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range is alsofrom 630 nm to 736 nm, to provide a wavelength bandwidth W of 106 nm.The center wavelength λ_(c) of the wavelength band is 683 nm. The valueobtained by dividing the wavelength bandwidth W by the center wavelengthλ_(c) is approximately 0.155, which is greater than 0.06. It may be saidthat the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range is a wide band.

Because the thicknesses of the first to eighth layer films 39 to 46 takeon the above-mentioned values, the optical thickness t of the coatingfilm 47, i.e. the sum of the products of the refractive indices andthicknesses of the respective layers of the coating film 47, is 1316.93nm. This value is approximately 7.70 times the value t_(r) (171 nm)which is a quarter of the center wavelength λ_(c). Thus, the coatingfilm 47 is a very thick film. This improves heat dissipationcharacteristics at the end surface 1 b of the semiconductor photonicelement 1 to suppress the increase in temperature of the end surface 1b.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 4% for the wavelength λ equal to λ_(c) (683 nm)when the coating film 2 having the refractive index n_(f) of “1.4989” asin the first preferred embodiment and a thickness d_(f) of five times113.92 nm, i.e. five times λ_(c)/(4n_(f)), is provided on the endsurface 1 b of the semiconductor photonic element 1. FIG. 32 shows thewavelength dependence of the power reflectivity R in this case. Asillustrated in FIG. 32, the wavelength band for which the powerreflectivity R falls within ±2% from the design reflectivity of 4% isfrom 663 nm to 705 nm, to provide a wavelength bandwidth W_(r) of 42 nm.

Thus, the wavelength bandwidth W (106 nm) for the coating film 47 of thesixth preferred embodiment is greater than the wavelength bandwidthW_(r) (42 nm) for the coating film 2 shown in FIG. 2.

Because the wavelength λ of light propagating through the active layer 1a is sometimes varied, it is desirable that the center wavelength λ_(c)of the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range be equal to or close to the designwavelength λ_(t). For example, when d₃=38.03 nm in the above instance,the basic amounts of phase change φ₁ and φ₂ are “0.674368” and“1.15312,” respectively. The center wavelength λ_(c) takes on a valueequal to the design wavelength λ_(t) of 650 nm when the thickness of thecoating film 47 is determined using the basic thicknesses d₁ and d₂obtained by substituting the above-mentioned values of the basic amountsof phase change φ₁ and φ₂ into Equations (5a) and (5b) and substituting618 nm, rather than the design wavelength λ_(t), for λ in Equations (5a)and (5b). FIG. 33 shows the wavelength dependence of the powerreflectivity R in this case.

As illustrated in FIG. 33, the power reflectivity R falls within a rangefrom 4.0% to 6.0% for the wavelength λ ranging from 599 nm to 700 nm.The wavelength band for which the power reflectivity R falls within ±2%from the design reflectivity of 4% is also from 599 nm to 700 nm, toprovide a wavelength bandwidth W of 101 nm. The center wavelength λ_(c)of the wavelength band is 650 nm, which is equal to the designwavelength of 650 nm. The value obtained by dividing the wavelengthbandwidth W (101 nm) by the center wavelength λ_(c) (650 nm) isapproximately 0.155, which is greater than 0.06 and shows that thewavelength band for which the power reflectivity R falls within theallowable reflectivity range is a wide band.

In this case, the thicknesses of the first to eighth layer films 39 to46 of the coating film 47 are 38.03 nm, 191.59 nm, 61.27 nm, 145.61 nm,32.25 nm, 76.63 nm, 66.10 nm and 157.10 nm, respectively. The opticalthickness t of the coating film 47 is 1252.11 nm. This value isapproximately 7.68 times the value t_(r) (163 nm) which is a quarter ofthe center wavelength λ_(c). Thus, the coating film 47 is a very thickfilm.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 4% for the wavelength λ equal to λ_(c) (650 nm)when the coating film 2 having the above-mentioned refractive indexn_(f) of “1.4989” and a thickness d_(f) of five times 108.41 nm, i.e.five times λ_(c)/(4n_(f)), is provided on the end surface 1 b of thesemiconductor photonic element 1. FIG. 34 shows the wavelengthdependence of the power reflectivity R in this case. As illustrated inFIG. 34, the wavelength band for which the power reflectivity R fallswithin ±2% from the design reflectivity of 4% is from 631 nm to 670 nm,to provide a wavelength bandwidth W_(r) of 39 nm.

Thus, even when the center wavelength λ_(c) is made equal to the designwavelength λ_(t), the wavelength bandwidth W (101 nm) for the coatingfilm 47 is greater than the wavelength bandwidth W_(r) (39 nm) for thecoating film 2 shown in FIG. 2.

Seventh Preferred Embodiment

FIG. 35 is a side view showing a structure of the semiconductor photonicdevice according to the seventh preferred embodiment of the presentinvention. A coating film 57 having a nine-layer structure is providedon the end surface 1 b of the semiconductor photonic element 1 in theseventh preferred embodiment.

As illustrated in FIG. 35, the coating film 57 according to the seventhpreferred embodiment of the present invention includes: a first layerfilm 48 having the refractive index n₂ and the thickness Od₂; a secondlayer film 49 having the refractive index n₁ and the thickness Ad₁; athird layer film 50 having the refractive index n₂ and the thicknessAd₂; a fourth layer film 51 having the refractive index n₁ and thethickness Bd₁; a fifth layer film 52 having the refractive index n₂ andthe thickness Bd₂; a sixth layer film 53 having the refractive index n₁and the thickness Cd₁; a seventh layer film 54 having the refractiveindex n₂ and the thickness Cd₂, an eighth layer film 55 having therefractive index n₁ and the thickness Dd₁; and a ninth layer film 56having the refractive index n₂ and the thickness Dd₂.

Each of the first layer film 48, the third layer film 50, the fifthlayer film 52, the seventh layer film 54 and the ninth layer film 56 isan alumina layer, and each of the second layer film 49, the fourth layerfilm 51, the sixth layer film 53 and the eighth layer film 55 is atantalum oxide layer. Thus, the coating film 57 according to the seventhpreferred embodiment is composed of two material layers: the aluminalayer and the tantalum oxide layer.

In the seventh preferred embodiment, the second and third layer films 49and 50, the fourth and fifth layer films 51 and 52, the sixth andseventh layer films 53 and 54, and the eighth and ninth layer films 55and 56 constitute unit layer pairs each composed of the alumina layerand the tantalum oxide layer arranged in a stacked relation. Thereference characters A, B, C and D which determine the thicknesses ofthe second to ninth layer films 49 to 56 of the coating film 57designate parameters individually determined for the respective unitlayer pairs and each indicating a contribution ratio of the thickness ofa corresponding unit layer pair to the thickness of the entire coatingfilm 57. The reference character O which determines the thickness of thefirst layer film 48 designates a parameter indicating a contributionratio of the thickness of the first layer film 48 to the thickness ofthe entire coating film 57. The reference characters d₁ and d₂ accordingto the seventh preferred embodiment also designate basic thicknessesindividually determined for the respective material layers.

The amounts of phase change for the first to ninth layer films 48 to 56are designated by Oφ₂, Aφ₁, Aφ₂, Bφ₁, Bφ₂, Cφ₁, Cφ₂, Dφ₁ and Dφ₂,respectively, using Equations (5a) and (5b). Therefore, the elementsm₁₁, m₁₂, m₂₁ and m₂₂ of a characteristic matrix for the coating film 57according to the seventh preferred embodiment satisfy the followingdeterminant: $\begin{matrix}{\begin{bmatrix}m_{11} & m_{12} \\m_{21} & m_{22}\end{bmatrix} = {\begin{bmatrix}{\cos\quad O\quad\phi_{2}} & {{- \frac{i}{n_{2}}}\sin\quad O\quad\phi_{2}} \\{{- {in}_{2}}\sin\quad O\quad\phi_{2}} & {\cos\quad O\quad\phi_{2}}\end{bmatrix} \times {\quad{\begin{bmatrix}{\cos\quad A\quad\phi_{1`}} & {{- \frac{i}{n_{1}}}\sin\quad A\quad\phi_{1}} \\{{- {in}_{1}}\sin\quad A\quad\phi_{1}} & {\cos\quad A\quad\phi_{1}}\end{bmatrix}\quad{\quad{\begin{bmatrix}{\cos\quad A\quad\phi_{2}} & {{- \frac{i}{n_{2}}}\sin\quad A\quad\phi_{2}} \\{{- {in}_{2}}\sin\quad A\quad\phi_{2}} & {\cos\quad A\quad\phi_{2}}\end{bmatrix} \times {\quad{\begin{bmatrix}{\cos\quad B\quad\phi_{1}} & {{- \frac{i}{n_{1}}}\sin\quad B\quad\phi_{1}} \\{{- {in}_{1}}\sin\quad B\quad\phi_{1}} & {\cos\quad B\quad\phi_{1}}\end{bmatrix}{\quad{\begin{bmatrix}{\cos\quad B\quad\phi_{2}} & {{- \frac{i}{n_{2}}}\sin\quad B\quad\phi_{2}} \\{{- {in}_{2}}\sin\quad B\quad\phi_{2}} & {\cos\quad B\quad\phi_{2}}\end{bmatrix} \times {\quad{\begin{bmatrix}{\cos\quad C\quad\phi_{1}} & {{- \frac{i}{n_{1}}}\sin\quad C\quad\phi_{1}} \\{{- {in}_{1}}\sin\quad C\quad\phi_{1}} & {\cos\quad C\quad\phi_{1}}\end{bmatrix}{\quad{\begin{bmatrix}{\cos\quad C\quad\phi_{2}} & {{- \frac{i}{n_{2}}}\sin\quad C\quad\phi_{2}} \\{{- {in}_{2}}\sin\quad C\quad\phi_{2}} & {\cos\quad C\quad\phi_{2}}\end{bmatrix} \times {\quad{\begin{bmatrix}{\cos\quad D\quad\phi_{1}} & {{- \frac{i}{n_{1}}}\sin\quad D\quad\phi_{1}} \\{{- {in}_{1}}\sin\quad D\quad\phi_{1}} & {\cos\quad D\quad\phi_{1}}\end{bmatrix}{\quad\begin{bmatrix}{\cos\quad D\quad\phi_{2}} & {{- \frac{i}{n_{2}}}\sin\quad D\quad\phi_{2}} \\{{- {in}_{2}}\sin\quad D\quad\phi_{2}} & {\cos\quad D\quad\phi_{2}}\end{bmatrix}}}}}}}}}}}}}}}}}} & \left( {7g} \right)\end{matrix}$

For designing the thickness of the coating film 57 so that the value ofthe amplitude reflectivity r is an imaginary number, the seventhpreferred embodiment previously determines the values of the parametersA, B, C, D and O, and executes Steps s1 to s5 using Equations (6) and(7g) described above to determine the values of the basic amounts ofphase change φ₁ and φ₂. Thereafter, Step s6 is executed to determine thevalues of the basic thicknesses d₁ and d₂, and the thicknesses of therespective layers of the coating film 57 are determined using thepreviously determined values of the parameters A, B, C, D and O and thevalues of the basic thicknesses d₁ and d₂ determined in Step s6. If thedesigned characteristic of the coating film 57 is insufficient, thebasic thicknesses d₁ and d₂ are determined again by changing theparameters A, B, C, D and O, and the thickness of the coating film 57 isdesigned again.

In the seventh preferred embodiment, the effective refractive indexn_(c) of the semiconductor photonic element 1 is “3.37.” The refractiveindices n₁ and n₂ are “2.057” and “1.620,” respectively. The designwavelength λ_(t) is 980 nm.

For designing the thickness of the coating film 57 having the powerreflectivity R of 4% (R_(t)=4%) when the wavelength λ equals the designwavelength of 980 nm under the above-mentioned conditions, a point whichprovides a phase angle θ of 240 degrees is selected, for example, inStep s1 so that the reflection amplitude vector rv is located in thethird quadrant (or the lower left quadrant) of the complex plane. Then,the values of the real and imaginary parts r_(r) and r_(i) of thecomplex number inputted as the value of the amplitude reflectivity r are“−0.1” and “−0.17320508,” respectively, in Step s2.

When O=0.46, A=1.44, B=2.00, C=2.00 and D=2.00 are set, the basicamounts of phase change φ₁ and φ₂ determined in Step s5 are “1.15080”and “0.506897,” respectively. Accordingly, the thicknesses of the firstto ninth layer films 48 to 56 determined in Step s6 are 22.45 nm, 125.65nm, 70.28 nm, 174.52 nm, 97.61 nm, 174.52 nm, 97.61 nm, 174.52 nm and97.61 nm, respectively.

FIG. 36 shows the wavelength dependence of the power reflectivity R ofthe coating film 57 thus designed. As illustrated in FIG. 36, the powerreflectivity R is 4% when the wavelength λ equals the design wavelengthof 980 nm. A wavelength band for which the power reflectivity R isapproximately equal to the design reflectivity of 4% is wide. The powerreflectivity R falls within a range from 4.0% to 6.0% for the wavelengthλ ranging from 913 nm to 1031 nm.

When the allowable reflectivity range is, for example, ±2% from thedesign reflectivity of 4%, the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range is alsofrom 913 nm to 1031 nm, to provide a wavelength bandwidth W of 118 nm.The center wavelength λ_(c) of the wavelength band is 972 nm. The valueobtained by dividing the wavelength bandwidth W by the center wavelengthλ_(c) is approximately 0.121, which is greater than 0.06. It may be saidthat the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range is a wide band.

Because the thicknesses of the first to ninth layer films 48 to 56 takeon the above-mentioned values, the optical thickness t of the coatingfilm 57, i.e. the sum of the products of the refractive indices andthicknesses of the respective layers of the coating film 57, is 1960.03nm. This value is approximately 8.07 times the value t_(r) (243 nm)which is a quarter of the center wavelength λ_(c). Thus, the coatingfilm 57 is a very thick film. This improves heat dissipationcharacteristics at the end surface 1 b of the semiconductor photonicelement 1 to suppress the increase in temperature of the end surface 1b.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 4% for the wavelength λ equal to λ_(c) (972 nm)when the coating film 2 having the refractive index n_(f) of “1.4989” asin the first preferred embodiment and a thickness d_(f) of five times162.12 nm, i.e. five times λ_(c)/(4n_(f)), is provided on the endsurface 1 b of the semiconductor photonic element 1. FIG. 37 shows thewavelength dependence of the power reflectivity R in this case. Asillustrated in FIG. 37, the wavelength band for which the powerreflectivity R falls within ±2% from the design reflectivity of 4% isfrom 943 nm to 1003 nm, to provide a wavelength bandwidth W_(r) of 60nm.

Thus, the wavelength bandwidth W (118 nm) for the coating film 57 of theseventh preferred embodiment is greater than the wavelength bandwidthW_(r) (60 nm) for the coating film 2 shown in FIG. 2.

Because the wavelength λ of light propagating through the active layer 1a is sometimes varied, it is desirable that the center wavelength λ_(c)of the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range be equal to or close to the designwavelength λ_(t). For example, the center wavelength λ_(c) takes on avalue equal to the design wavelength λ_(t) of 980 nm when the thicknessof the coating film 57 is determined using the basic thicknesses d₁ andd₂ obtained by setting the basic amounts of phase change φ₁ and φ₂ at“1.15080” and “0.506897,” respectively, in a similar manner to the aboveinstance and substituting 988 nm, rather than the design wavelengthλ_(t), for λ in Equations (5a) and (5b). FIG. 38 shows the wavelengthdependence of the power reflectivity R in this case.

As illustrated in FIG. 38, the power reflectivity R falls within a rangefrom 4.0% to 6.0% for the wavelength λ ranging from 921 nm to 1039 nm.The wavelength band for which the power reflectivity R falls within ±2%from the design reflectivity of 4% is also from 921 nm to 1039 nm, toprovide a wavelength bandwidth W of 118 nm. The center wavelength λ_(c)of the wavelength band is 980 nm, which is equal to the designwavelength of 980 nm. The value obtained by dividing the wavelengthbandwidth W (118 nm) by the center wavelength λ_(c) (980 nm) isapproximately 0.120, which is greater than 0.06 and shows that thewavelength band for which the power reflectivity R falls within theallowable reflectivity range is a wide band.

In this case, the thicknesses of the first to ninth layer films 48 to 56of the coating film 57 are 22.63 nm, 126.68 nm, 70.85 nm, 175.94 nm,98.40 nm, 175.94 nm, 98.40 nm, 175.94 nm and 98.40 nm, respectively. Theoptical thickness t of the coating film 57 is 1975.97 nm. This value isapproximately 8.07 times the value t_(r) (245 nm) which is a quarter ofthe center wavelength λ_(c). Thus, the coating film 57 is a very thickfilm.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 4% for the wavelength λ equal to λ_(c) (980 nm)when the coating film 2 having the above-mentioned refractive indexn_(f) of “1.4989” and a thickness d_(f) of five times 163.45 nm, i.e.five times λ_(c)/(4n_(f)), is provided on the end surface 1 b of thesemiconductor photonic element 1. The wavelength dependence of the powerreflectivity R in this case is substantially similar to that shown inFIG. 10 described above. As illustrated in FIG. 10, the wavelength bandfor which the power reflectivity R falls within ±2% from the designreflectivity of 4% is from 951 nm to 1011 nm, to provide a wavelengthbandwidth W_(r) of 60 nm.

Thus, even when the center wavelength λ_(c) is made equal to the designwavelength λ_(t), the wavelength bandwidth W (118 nm) for the coatingfilm 57 is greater than the wavelength bandwidth W_(r) (60 nm) for thecoating film 2 shown in FIG. 2.

Eighth Preferred Embodiment

FIG. 39 is a side view showing a structure of the semiconductor photonicdevice according to the eighth preferred embodiment of the presentinvention. A coating film 68 having a ten-layer structure is provided onthe end surface 1 b of the semiconductor photonic element 1 in theeighth preferred embodiment.

As illustrated in FIG. 39, the coating film 68 according to the eighthpreferred embodiment of the present invention includes: a first layerfilm 58 having the refractive index n₃ and the thickness d₃; a secondlayer film 59 having the refractive index n₂ and the thickness Ad₂; athird layer film 60 having the refractive index n₁ and the thicknessBd₁; a fourth layer film 61 having the refractive index n₂ and thethickness Bd₂; a fifth layer film 62 having the refractive index n₁ andthe thickness Cd₁; a sixth layer film 63 having the refractive index n₂and the thickness Cd₂; a seventh layer film 64 having the refractiveindex n₁ and the thickness Dd₁; an eighth layer film 65 having therefractive index n₂ and the thickness Dd₂; a ninth layer film 66 havingthe refractive index n₁ and a thickness Ed₁; and a tenth layer film 67having the refractive index n₂ and a thickness Ed₂.

The first layer film 58 is an alumina layer. Each of the second layerfilm 59, the fourth layer film 61, the sixth layer film 63, the eighthlayer film 65 and the tenth layer film 67 is a silicon oxide layer. Eachof the third layer film 60, the fifth layer film 62, the seventh layerfilm 64 and the ninth layer film 66 is an aluminum nitride layer. Thus,the coating film 68 according to the eighth preferred embodiment iscomposed of three material layers: the alumina layer, the silicon oxidelayer, and the aluminum nitride layer.

In the eighth preferred embodiment, the third and fourth layer films 60and 61, the fifth and sixth layer films 62 and 63, the seventh andeighth layer films 64 and 65, and the ninth and tenth layer films 66 and67 constitute unit layer pairs each composed of the silicon oxide layerand the aluminum nitride layer arranged in a stacked relation. Thereference characters B, C, D and E which determine the thicknesses ofthe third to tenth layer films 60 to 67 of the coating film 68 designateparameters individually determined for the respective unit layer pairsand each indicating a contribution ratio of the thickness of acorresponding unit layer pair to the thickness of the entire coatingfilm 68. The reference character A which determines the thickness of thesecond layer film 59 designates a parameter indicating a contributionratio of the thickness of the second layer film 59 to the thickness ofthe entire coating film 68. The reference characters d₁ and d₂ accordingto the eighth preferred embodiment also designate basic thicknessesindividually determined for the respective material layers.

The amounts of phase change for the second to tenth layer films 59 to 67are designated by Aφ₂, Bφ₁, Bφ₂, Cφ₁, Cφ₂, Dφ₁, Dφ₂, Eφ₁ and Eφ₂,respectively, using Equations (5a) and (5b). The amount of phase changeφ₃ for the first layer film 58 is expressed by Equation (9) describedabove. Therefore, the elements m₁₁, m₁₂, m₂₁ and m₂₂ of a characteristicmatrix for the coating film 68 according to the eighth preferredembodiment satisfy the following determinant: $\begin{matrix}{\begin{bmatrix}m_{11} & m_{12} \\m_{21} & m_{22}\end{bmatrix} = {\begin{bmatrix}{\cos\quad\phi_{3}} & {{- \frac{i}{n_{3}}}\sin\quad\phi_{3}} \\{{- {in}_{3}}\sin\quad\phi_{3}} & {\cos\quad\phi_{3}}\end{bmatrix}{\quad{\begin{bmatrix}{\cos\quad A\quad\phi_{2}} & {{- \frac{i}{n_{2}}}\sin\quad A\quad\phi_{2}} \\{{- {in}_{2}}\sin\quad A\quad\phi_{2}} & {\cos\quad A\quad\phi_{2}}\end{bmatrix} \times \begin{bmatrix}{\cos\quad B\quad\phi_{1}} & {{- \frac{i}{n_{1}}}\sin\quad B\quad\phi_{1}} \\{{- {in}_{1}}\sin\quad B\quad\phi_{1}} & {\cos\quad B\quad\phi_{1}}\end{bmatrix}{\quad{\begin{bmatrix}{\cos\quad B\quad\phi_{2}} & {{- \frac{i}{n_{2}}}\sin\quad B\quad\phi_{2}} \\{{- {in}_{2}}\sin\quad B\quad\phi_{2}} & {\cos\quad B\quad\phi_{2}}\end{bmatrix} \times \begin{bmatrix}{\cos\quad C\quad\phi_{1}} & {{- \frac{i}{n_{1}}}\sin\quad C\quad\phi_{1}} \\{{- {in}_{1}}\sin\quad C\quad\phi_{1}} & {\cos\quad C\quad\phi_{1}}\end{bmatrix}{\quad{\begin{bmatrix}{\cos\quad C\quad\phi_{2}} & {{- \frac{i}{n_{2}}}\sin\quad C\quad\phi_{2}} \\{{- {in}_{2}}\sin\quad C\quad\phi_{2}} & {\cos\quad C\quad\phi_{2}}\end{bmatrix} \times {\quad{\begin{bmatrix}{\cos\quad D\quad\phi_{1}} & {{- \frac{i}{n_{1}}}\sin\quad D\quad\phi_{1}} \\{{- {in}_{1}}\sin\quad D\quad\phi_{1}} & {\cos\quad D\quad\phi_{1}}\end{bmatrix}{\quad{\begin{bmatrix}{\cos\quad D\quad\phi_{2}} & {{- \frac{i}{n_{2}}}\sin\quad D\quad\phi_{2}} \\{{- {in}_{2}}\sin\quad D\quad\phi_{2}} & {\cos\quad D\quad\phi_{2}}\end{bmatrix} \times {\quad{\begin{bmatrix}{\cos\quad E\quad\phi_{1}} & {{- \frac{i}{n_{1}}}\sin\quad E\quad\phi_{1}} \\{{- {in}_{1}}\sin\quad E\quad\phi_{1}} & {\cos\quad E\quad\phi_{1}}\end{bmatrix}{\quad\begin{bmatrix}{\cos\quad E\quad\phi_{2}} & {{- \frac{i}{n_{2}}}\sin\quad E\quad\phi_{2}} \\{{- {in}_{2}}\sin\quad E\quad\phi_{2}} & {\cos\quad E\quad\phi_{2}}\end{bmatrix}}}}}}}}}}}}}}}} & \left( {7h} \right)\end{matrix}$

For designing the thickness of the coating film 68 so that the value ofthe amplitude reflectivity r is an imaginary number, the eighthpreferred embodiment previously determines the values of the parametersA, B, C, D and E, and previously determines the thickness d₃ of thefirst layer film 58, thereby to handle the value of the amount of phasechange φ₃ as a known value. Steps s1 to s5 are executed using Equations(6) and (7h) described above to determine the values of the basicamounts of phase change φ₁ and φ₂. Thereafter, Step s6 is executed todetermine the values of the basic thicknesses d₁ and d₂, and thethicknesses of the second to tenth layer films 59 to 67 of the coatingfilm 68 are determined using the previously determined values of theparameters A, B, C, D and E and the values of the basic thicknesses d₁and d₂ determined in Step s6. If the designed characteristic of thecoating film 68 is insufficient, the basic thicknesses d₁ and d₂ aredetermined again by changing the parameters A, B, C, D and E or thethickness d₃, and the thickness of the coating film 68 is designedagain.

In the eighth preferred embodiment, the effective refractive index n_(c)of the semiconductor photonic element 1 is “3.37.” The refractiveindices n₁ to n₃ are “2.072,” “1.480” and “1.620,” respectively. Thedesign wavelength λ_(t) is 808 nm.

For designing the thickness of the coating film 68 having the powerreflectivity R of 4% when the wavelength λ equals the design wavelengthof 808 nm under the above-mentioned conditions, a point which provides aphase angle θ of 315 degrees is selected, for example, in Step s1 sothat the reflection amplitude vector rv is located in the fourthquadrant (or the lower right quadrant) of the complex plane. Then, thevalues of the real and imaginary parts r_(r) and r_(i) of the complexnumber inputted as the value of the amplitude reflectivity r are“+0.141421356” and “−0.141421356,”, respectively, in Step s2.

When A=0.63, B=1.87, C=2.01, D=2.00, E=2.00 and d₃=40.0 nm are set, thebasic amounts of phase change φ₁ and φ₂ determined in Step s5 are“0.219827” and “1.23802,” respectively. Accordingly, the thicknesses ofthe second to tenth layer films 59 to 67 determined in Step s6 are 67.77nm, 25.51 nm, 201.16 nm, 27.42 nm, 216.22 nm, 27.29 nm, 215.14 nm, 27.29nm and 215.14 nm, respectively.

FIG. 40 shows the wavelength dependence of the power reflectivity R ofthe coating film 68 thus designed. As illustrated in FIG. 40, the powerreflectivity R is 4% when the wavelength λ equals the design wavelengthof 808 nm. A wavelength band for which the power reflectivity R isapproximately equal to the design reflectivity of 4% is wide. The powerreflectivity R falls within a range from 4.0% to 6.0% for the wavelengthλ ranging from 793 nm to 893 nm.

When the allowable reflectivity range is, for example, ±2% from thedesign reflectivity of 4%, the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range is alsofrom 793 nm to 893 nm, to provide a the wavelength bandwidth W of 100nm. The center wavelength λ_(c) of the wavelength band is 843 nm. Thevalue obtained by dividing the wavelength bandwidth W by the centerwavelength λ_(c) is approximately 0.119, which is greater than 0.06. Itmay be said that the wavelength band for which the power reflectivity Rfalls within the allowable reflectivity range is a wide band.

Because the thicknesses of the first to tenth layer films 58 to 67 takeon the above-mentioned values, the optical thickness t of the coatingfilm 68, i.e. the sum of the products of the refractive indices andthicknesses of the respective layers of the coating film 68, is 1642.40nm. This value is approximately 7.78 times the value t_(r) (211 nm)which is a quarter of the center wavelength λ_(c). Thus, the coatingfilm 68 is a very thick film. This improves heat dissipationcharacteristics at the end surface 1 b of the semiconductor photonicelement 1 to suppress the increase in temperature of the end surface 1b.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 4% for the wavelength λ equal to λ_(c) (843 nm)when the coating film 2 having the refractive index n_(f) of “1.4989” asin the first preferred embodiment and a thickness d_(f) of five times140.60 nm, i.e. five times λ_(c)/(4n_(f)), is provided on the endsurface 1 b of the semiconductor photonic element 1. FIG. 41 shows thewavelength dependence of the power reflectivity R in this case. Asillustrated in FIG. 41, the wavelength band for which the powerreflectivity R falls within ±2% from the design reflectivity of 4% isfrom 818 nm to 870 nm, to provide a wavelength bandwidth W_(r) of 52 nm.

Thus, the wavelength bandwidth W (100 nm) for the coating film 68 of theeighth preferred embodiment is greater than the wavelength bandwidthW_(r) (52 nm) for the coating film 2 shown in FIG. 2.

Because the wavelength λ of light propagating through the active layer 1a is sometimes varied, it is desirable that the center wavelength λ_(c)of the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range be equal to or close to the designwavelength λ_(t). For example, when d₃=40.47 nm, A=0.63, B=1.87, C=1.96,D=2.00 and E=2.00 are set, the basic amounts of phase change φ₁ and φ₂are “0.235529” and “1.21623,” respectively. The center wavelength λ_(c)takes on a value equal to the design wavelength of 808 nm when thethickness of the coating film 68 is determined using the basicthicknesses d₁ and d₂ obtained by substituting the above-mentionedvalues of the basic amounts of phase change φ₁ and φ₂ into Equations(5a) and (5b) and substituting 779 nm, rather than the design wavelengthλ_(t), for λ in Equations (5a) and (5b). FIG. 42 shows the wavelengthdependence of the power reflectivity R in this case.

As illustrated in FIG. 42, the power reflectivity R falls within a rangefrom 2.0% to 6.0% for the wavelength λ ranging from 763 nm to 853 nm.The wavelength band for which the power reflectivity R falls within ±2%from the design reflectivity of 4% is also from 763 nm to 853 nm, toprovide a wavelength bandwidth W of 90 nm. The center wavelength λ_(c)of the wavelength band is 808 nm, which is equal to the designwavelength of 808 nm. The value obtained by dividing the wavelengthbandwidth W (90 nm) by the center wavelength λ_(c) (808 nm) isapproximately 0.111, which is greater than 0.06 and shows that thewavelength band for which the power reflectivity R falls within theallowable reflectivity range is a wide band.

In this case, the thicknesses of the first to tenth layer films 58 to 67of the coating film 68 are 40.47 nm, 64.19 nm, 26.35 nm, 190.53 nm,27.62 nm, 199.70 nm, 28.19 nm, 203.77 nm, 28.19 nm and 203.77 nm,respectively. The optical thickness t of the coating film 68 is 1569.91nm. This value is approximately 7.77 times the value t_(r) (202 nm)which is a quarter of the center wavelength λ_(c). Thus, the coatingfilm 68 is a very thick film.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 4% for the wavelength λ equal to λ_(c) (808 nm)when the coating film 2 having the above-mentioned refractive indexn_(f) of “1.4989” and a thickness d_(f) of five times 134.77 nm, i.e.five times λ_(c)/(4n_(f)), is provided on the end surface 1 b of thesemiconductor photonic element 1. FIG. 43 shows the wavelengthdependence of the power reflectivity R in this case. As illustrated inFIG. 43, the wavelength band for which the power reflectivity R fallswithin ±2% from the design reflectivity of 4% is from 774 nm to 834 nm,to provide a wavelength bandwidth W_(r) of 60 nm.

Thus, even when the center wavelength λ_(c) is made equal to the designwavelength λ_(t), the wavelength bandwidth W (90 nm) for the coatingfilm 68 is greater than the wavelength bandwidth W_(r) (60 nm) for thecoating film 2 shown in FIG. 2.

Ninth Preferred Embodiment

FIG. 44 shows the wavelength dependence of the power reflectivity R ofthe coating film 13 in the semiconductor photonic device according tothe ninth preferred embodiment of the present invention. Thesemiconductor photonic device according to the ninth preferredembodiment is similar to the semiconductor photonic device according tothe first preferred embodiment except that the design wavelength λ_(t)is changed from 980 nm to 1310 nm.

For designing the thickness of the coating film 13 having a powerreflectivity R of 8% (R_(t)=8%) when the wavelength λ equals the designwavelength of 1310 nm, a point which provides a phase angle θ of 30degrees is selected, for example, in Step s1 so that the reflectionamplitude vector rv is located in the first quadrant (or the upper rightquadrant) of the complex plane according to the ninth preferredembodiment. Then, because the magnitude of the complex number at theselected point is 0.282842712, the values of the real and imaginaryparts r_(r) and r_(i) of the complex number inputted as the value of theamplitude reflectivity r are “+0.244948974” and “+0.141421356,”respectively, in Step s2.

When A=3.15, B=2.54 and C=2.05 are set, the basic amounts of phasechange φ₁ and φ₂ determined in Step s5 are “1.23348” and “0.560095,”respectively. Accordingly, the thicknesses of the first to sixth layerfilms 7 to 12 determined in Step s6 are 393.82 nm, 248.54 nm, 317.56 nm,200.41 nm, 256.30 nm and 161.75 nm, respectively.

FIG. 44 described above shows the wavelength dependence of the powerreflectivity R of the coating film 13 thus designed. As illustrated inFIG. 44, the power reflectivity R is 8% when the wavelength λ equals thedesign wavelength of 1310 nm. A wavelength band for which the powerreflectivity R is approximately equal to the design reflectivity of 8%is wide. The power reflectivity R falls within a range from 6.5% to10.0% for the wavelength λ ranging from 1059 nm to 1509 nm.

When the allowable reflectivity range is, for example, ±2% from thedesign reflectivity of 8%, the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range is alsofrom 1059 nm to 1509 nm, to provide a wavelength bandwidth W of 450 nm.The center wavelength λ_(c) of the wavelength band is 1284 nm. The valueobtained by dividing the wavelength bandwidth W by the center wavelengthλ_(c) is approximately 0.356, which is greater than 0.06. It may be saidthat the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range is a wide band.

Because the thicknesses of the first to sixth layer films 7 to 12 takeon the above-mentioned values, the optical thickness t of the coatingfilm 13, i.e. the sum of the products of the refractive indices andthicknesses of the respective layers of the coating film 13, is 2894.35nm. This value is approximately 9.02 times the value t_(r) (321 nm)which is a quarter of the center wavelength λ_(c). Thus, the coatingfilm 13 is a very thick film. This improves heat dissipationcharacteristics at the end surface 1 b of the semiconductor photonicelement 1 to suppress the increase in temperature of the end surface 1b.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 8% for the wavelength λ equal to λ_(c) (1284 nm)when the coating film 2 having a refractive index n_(f) of “1.3726” anda thickness d_(f) of five times 223.86 nm, i.e. five timesλ_(c)/(4n_(f)), is provided on the end surface 1 b of the semiconductorphotonic element 1. FIG. 45 shows the wavelength dependence of the powerreflectivity R in this case. As illustrated in FIG. 45, the wavelengthband for which the power reflectivity R falls within ±2% from the designreflectivity of 8% is from 1241 nm to 1330 nm, to provide a wavelengthbandwidth W_(r) of 89 nm.

The above-mentioned value “1.3726” of the refractive index n_(f) isobtained by substituting R_(t)=0.08 and n_(c)=3.37 into Equation (8).

Thus, the wavelength bandwidth W (450 nm) for the coating film 13 of theninth preferred embodiment is greater than the wavelength bandwidthW_(r) (89 nm) for the coating film 2 shown in FIG. 2.

As mentioned above, because the wavelength λ of light propagatingthrough the active layer 1 a is sometimes varied, it is desirable thatthe center wavelength λ_(c) of the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range be equal toor close to the design wavelength λ_(t). For example, the centerwavelength λ_(c) takes on a value equal to the design wavelength λ_(t)of 1310 nm when the thickness of the coating film 13 is determined usingthe basic thicknesses d₁ and d₂ obtained by setting the basic amounts ofphase change φ₁ and φ₂ at “1.23348” and “0.560095,” respectively, in asimilar manner to the above instance and substituting 1336 nm, ratherthan the design wavelength λ_(t), for λ in Equations (5a) and (5b). FIG.46 shows the wavelength dependence of the power reflectivity R in thiscase.

As illustrated in FIG. 46, the power reflectivity R falls within a rangefrom 6.5% to 10.0% for the wavelength λ ranging from 1080 nm to 1539 nm.The wavelength band for which the power reflectivity R falls within ±2%from the design reflectivity of 8% is also from 1080 nm to 1539 nm, toprovide a wavelength bandwidth W of 459 nm. The center wavelength λ_(c)of the wavelength band is 1310 nm, which is equal to the designwavelength of 1310 nm. The value obtained by dividing the wavelengthbandwidth W (459 nm) by the center wavelength λ_(c) (1310 nm) isapproximately 0.350, which is greater than 0.06 and shows that thewavelength band for which the power reflectivity R falls within theallowable reflectivity range is a wide band.

In this case, the thicknesses of the first to sixth layer films 7 to 12of the coating film 13 are 401.64 nm, 253.48 nm, 323.86 nm, 204.39 nm,261.38 nm and 164.96 nm, respectively. The optical thickness t of thecoating film 13 is 2951.80 nm. This value is approximately 9.00 timesthe value t_(r) (328 nm) which is a quarter of the center wavelengthλ_(c). Thus, the coating film 13 is a very thick film.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 8% for the wavelength λ equal to λ_(c) (1310 nm)when the coating film 2 having the above-mentioned refractive indexn_(f) of “1.3726” and a thickness d_(f) of five times 238.60 nm, i.e.five times λ_(c)/(4n_(f)), is provided on the end surface 1 b of thesemiconductor photonic element 1. FIG. 47 shows the wavelengthdependence of the power reflectivity R in this case. As illustrated inFIG. 47, the wavelength band for which the power reflectivity R fallswithin ±2% from the design reflectivity of 8% is from 1266 nm to 1357nm, to provide a wavelength bandwidth W_(r) of 91 nm.

Thus, even when the center wavelength λ_(c) is made equal to the designwavelength λ_(t), the wavelength bandwidth W (459 nm) for the coatingfilm 13 is greater than the wavelength bandwidth W_(r) (91 nm) for thecoating film 2 shown in FIG. 2.

Tenth Preferred Embodiment

FIG. 48 shows the wavelength dependence of the power reflectivity R ofthe coating film 13 in the semiconductor photonic device according tothe tenth preferred embodiment of the present invention. Thesemiconductor photonic device according to the tenth preferredembodiment is similar to the semiconductor photonic device according tothe second preferred embodiment except that the design wavelength λ_(t)is changed from 808 nm to 1550 nm.

For designing the thickness of the coating film 13 having the powerreflectivity R of 8% (R_(t)=8%) when the wavelength λ equals the designwavelength of 1550 nm, a point which provides a phase angle θ of 120degrees is selected, for example, in Step s1 so that the reflectionamplitude vector rv is located in the second quadrant (or the upper leftquadrant) of the complex plane according to the tenth preferredembodiment. Then, the values of the real and imaginary parts r_(r) andr_(i) of the complex number inputted as the value of the amplitudereflectivity r are “−0.141421356” and “+0.244948974,” respectively, inStep s2.

When A=2.00, B=2.00 and C=2.00 are set, the basic amounts of phasechange φ₁ and φ₂ determined in Step s5 are “0.591234” and “1.06568,”respectively. Accordingly, the thicknesses of the first to sixth layerfilms 7 to 12 determined in Step s6 are 141.81 nm, 324.56 nm, 141.81 nm,324.56 nm, 141.81 nm and 324.56 nm, respectively.

FIG. 48 described above shows the wavelength dependence of the powerreflectivity R of the coating film 13 thus designed. As illustrated inFIG. 48, the power reflectivity R is 8% when the wavelength λ equals thedesign wavelength of 1550 nm. A wavelength band for which the powerreflectivity R is approximately equal to the design reflectivity of 8%is wide. The power reflectivity R falls within a range from 7.2% to10.0% for the wavelength λ ranging from 1441 nm to 1868 nm.

When the allowable reflectivity range is, for example, ±2% from thedesign reflectivity of 8%, the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range is alsofrom 1441 nm to 1868 nm, to provide a wavelength bandwidth W of 427 nm.The center wavelength λ_(c) of the wavelength band is 1655 nm. The valueobtained by dividing the wavelength bandwidth W by the center wavelengthλ_(c) is approximately 0.258, which is greater than 0.06. It may be saidthat the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range is a wide band.

Because the thicknesses of the first to sixth layer films 7 to 12 takeon the above-mentioned values, the optical thickness t of the coatingfilm 13 is 2452.47 nm. This value is approximately 5.92 times the valuet_(r) (414 nm) which is a quarter of the center wavelength λ_(c). Thus,the coating film 13 is a very thick film. This improves heat dissipationcharacteristics at the end surface 1 b of the semiconductor photonicelement 1 to suppress the increase in temperature of the end surface 1b.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 8% for the wavelength λ equal to λ_(c) (1655 nm)when the coating film 2 having the refractive index n_(f) of “1.3726” asin the ninth preferred embodiment and a thickness d_(f) of five times301.44 nm, i.e. five times λ_(c)/(4n_(f)), is provided on the endsurface 1 b of the semiconductor photonic element 1. FIG. 49 shows thewavelength dependence of the power reflectivity R in this case. Asillustrated in FIG. 49, the wavelength band for which the powerreflectivity R falls within ±2% from the design reflectivity of 8% isfrom 1600 nm to 1714 nm, to provide a wavelength bandwidth W_(r) of 114nm.

Thus, the wavelength bandwidth W (427 nm) for the coating film 13 of thetenth preferred embodiment is wider than the wavelength bandwidth W_(r)(114 nm) for the coating film 2 shown in FIG. 2.

As mentioned above, because the wavelength λ of light propagatingthrough the active layer 1 a is sometimes varied, it is desirable thatthe center wavelength λ_(c) of the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range be equal toor close to the design wavelength λ_(t). For example, the centerwavelength λ_(c) takes on a value equal to the design wavelength λ_(t)of 1510 nm when the thickness of the coating film 13 is determined usingthe basic thicknesses d₁ and d₂ obtained by setting the basic amounts ofphase change φ₁ and φ₂ at “0.591234” and “1.06568,” respectively, in asimilar manner to the above instance and substituting 1452 nm, ratherthan the design wavelength λ_(t), for λ in Equations (5a) and (5b). FIG.50 shows the wavelength dependence of the power reflectivity R in thiscase.

As illustrated in FIG. 50, the power reflectivity R falls within a rangefrom 7.2% to 10.0% for the wavelength λ ranging from 1350 nm to 1750 nm.The wavelength band for which the power reflectivity R falls within ±2%from the design reflectivity of 8% is also from 1350 nm to 1750 nm, toprovide a wavelength bandwidth W of 400 nm. The center wavelength λ_(c)of the wavelength band is 1550 nm, which is equal to the designwavelength of 1550 nm. The value obtained by dividing the wavelengthbandwidth W (400 nm) by the center wavelength λ_(c) (1550 nm) isapproximately 0.258, which is greater than 0.06 and shows that thewavelength band for which the power reflectivity R falls within theallowable reflectivity range is a wide band.

In this case, the thicknesses of the first to sixth layer films 7 to 12of the coating film 13 are 132.84 nm, 304.04 nm, 132.84 nm, 304.04 nm,132.84 nm and 304.04 nm, respectively. The optical thickness t of thecoating film 13 is 2297.39 nm. This value is approximately 5.92 timesthe value t_(r) (388 nm) which is a quarter of the center wavelengthλ_(c). Thus, the coating film 13 is a very thick film.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 8% for the wavelength λ equal to λ_(c) (1550 nm)when the coating film 2 having the above-mentioned refractive indexn_(f) of “1.3726” and a thickness d_(f) of five times 282.31 nm, i.e.five times λ_(c)/(4n_(f)), is provided on the end surface 1 b of thesemiconductor photonic element 1. FIG. 51 shows the wavelengthdependence of the power reflectivity R in this case. As illustrated inFIG. 51, the wavelength band for which the power reflectivity R fallswithin ±2% from the design reflectivity of 8% is from 1498 nm to 1606nm, to provide a wavelength bandwidth W_(r) of 108 nm.

Thus, even when the center wavelength λ_(c) is made equal to the designwavelength λ_(t), the wavelength bandwidth W (400 nm) for the coatingfilm 13 is greater than the wavelength bandwidth W_(r) (108 nm) for thecoating film 2 shown in FIG. 2.

Eleventh Preferred Embodiment

FIG. 52 shows the wavelength dependence of the power reflectivity R ofthe coating film 21 in the semiconductor photonic device according tothe eleventh preferred embodiment of the present invention. Thesemiconductor photonic device according to the eleventh preferredembodiment is similar to the semiconductor photonic device shown in FIG.15 according to the third preferred embodiment except that the materiallayer for the first layer film 14, the third layer film 16, the fifthlayer film 18 and the seventh layer film 20 is changed from the aluminalayer to a silicon oxide layer, that the effective refractive indexn_(c) of the semiconductor photonic element 1 is changed from “3.37” to“2.50,” and that the design wavelength λ_(t) is changed from 1310 nm to410 nm.

Thus, because the silicon oxide layer is employed in place of thealumina layer, the refractive index n₂ of the first layer film 14, thethird layer film 16, the fifth layer film 18 and the seventh layer film20 is 1.480 according to the eleventh preferred embodiment. Because thedesign wavelength λ_(t) is 410 nm, the refractive index n₁ of the secondlayer film 15, the fourth layer film 17 and the sixth layer film 19which are the tantalum oxide layers is set at “2.128” in considerationfor wavelength dispersion.

Because GaN based semiconductor is employed in the semiconductorphotonic element 1 of the eleventh preferred embodiment, the effectiverefractive index n_(c) of the semiconductor photonic element 1 is“2.50,” as described above.

For designing the thickness of the coating film 21 having the powerreflectivity R of 8% (R_(t)=8%) when the wavelength λ equals the designwavelength of 410 nm, a point which provides a phase angle θ of 210degrees is selected, for example, in Step s1 so that the reflectionamplitude vector rv is located in the third quadrant (or the lower leftquadrant) of the complex plane according to the eleventh preferredembodiment. Then, the values of the real and imaginary parts r_(r) andr_(i) of the complex number inputted as the value of the amplitudereflectivity r are “−0.244948974” and “−0.141421356,” respectively, inStep s2.

When O=0.25, A=2.35, B=2.00 and C=2.00 are set, the basic amounts ofphase change φ₁ and φ₂ determined in Step s5 are “1.98646” and“0.294825,” respectively. Accordingly, the thicknesses of the first toseventh layer films 14 to 20 determined in Step s6 are 3.25 nm, 143.15nm, 30.55 nm, 121.83 nm, 26.00 nm, 121.83 nm and 26.00 nm, respectively.

FIG. 52 described above shows the wavelength dependence of the powerreflectivity R of the coating film 21 thus designed. As illustrated inFIG. 52, the power reflectivity R is 8% when the wavelength λ equals thedesign wavelength of 410 nm. A wavelength band for which the powerreflectivity R is approximately equal to the design reflectivity of 8%is wide. The power reflectivity R falls within a range from 7.7% to10.0% for the wavelength λ ranging from 362 nm to 424 nm.

When the allowable reflectivity range is, for example, ±2% from thedesign reflectivity of 8%, the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range is alsofrom 362 nm to 424 nm, to provide a wavelength bandwidth W of 62 nm. Thecenter wavelength λ_(c) of the wavelength band is 393 nm. The valueobtained by dividing the wavelength bandwidth W by the center wavelengthλ_(c) is approximately 0.158, which is greater than 0.06. It may be saidthat the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range is a wide band.

Because the thicknesses of the first to seventh layer films 14 to 20take on the above-mentioned values, the optical thickness t of thecoating film 21 is 950.12 nm. This value is approximately 9.70 times thevalue t_(r) (98 nm) which is a quarter of the center wavelength λ_(c).Thus, the coating film 21 is a very thick film. This improves heatdissipation characteristics at the end surface 1 b of the semiconductorphotonic element 1 to suppress the increase in temperature of the endsurface 1 b.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 8% for the wavelength λ equal to λ_(c) (393 nm)when the coating film 2 having a refractive index n_(f) of “1.1822” anda thickness d_(f) of five times 83.11 nm, i.e. five timesλ_(c)/(4n_(f)), is provided on the end surface 1 b of the semiconductorphotonic element 1. FIG. 53 shows the wavelength dependence of the powerreflectivity R in this case. As illustrated in FIG. 53, the wavelengthband for which the power reflectivity R falls within ±2% from the designreflectivity of 8% is from 373 nm to 415 nm, to provide a wavelengthbandwidth W_(r) of 42 nm.

The above-mentioned value “1.1822” of the refractive index n_(f) isobtained by substituting R_(t)=0.08 and n_(c)=2.50 into Equation (8).

Thus, the wavelength bandwidth W (62 nm) for the coating film 21 of theeleventh preferred embodiment is greater than the wavelength bandwidthW_(r) (42 nm) for the coating film 2 shown in FIG. 2.

As mentioned above, because the wavelength λ of light propagatingthrough the active layer 1 a is sometimes varied, it is desirable thatthe center wavelength λ_(c) of the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range be equal toor close to the design wavelength λ_(t). For example, the centerwavelength λ_(c) takes on a value equal to the design wavelength λ_(t)of 410 nm when the thickness of the coating film 21 is determined usingthe basic thicknesses d₁ and d₂ obtained by setting the basic amounts ofphase change φ₁ and φ₂ at “1.98646” and “0.294825,” respectively, in asimilar manner to the above instance and substituting 427 nm, ratherthan the design wavelength λ_(t), for λ in Equations (5a) and (5b). FIG.54 shows the wavelength dependence of the power reflectivity R in thiscase.

As illustrated in FIG. 54, the power reflectivity R falls within a rangefrom 7.7% to 10.0% for the wavelength λ ranging from 377 nm to 442 nm.The wavelength band for which the power reflectivity R falls within ±2%from the design reflectivity of 8% is also from 377 nm to 442 nm, toprovide a wavelength bandwidth W of 65 nm. The center wavelength λ_(c)of the wavelength band is 410 nm, which is equal to the designwavelength of 410 nm. The value obtained by dividing the wavelengthbandwidth W (65 nm) by the center wavelength λ_(c) (410 nm) isapproximately 0.159, which is greater than 0.06 and shows that thewavelength band for which the power reflectivity R falls within theallowable reflectivity range is a wide band.

In this case, the thicknesses of the first to seventh layer films 14 to20 of the coating film 21 are 3.38 nm, 149.08 nm, 31.81 nm, 126.88 nm,27.08 nm, 126.88 nm and 27.08 nm, respectively. The optical thickness tof the coating film 21 is 989.48 nm. This value is approximately 9.61times the value t_(r) (103 nm) which is a quarter of the centerwavelength λ_(c). Thus, the coating film 21 is a very thick film.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 8% for the wavelength λ equal to λ_(c) (410 nm)when the coating film 2 having the above-mentioned refractive indexn_(f) of “1.1822” and a thickness d_(f) of five times 86.70 nm, i.e.five times λ_(c)/(4n_(f)), is provided on the end surface 1 b of thesemiconductor photonic element 1. FIG. 55 shows the wavelengthdependence of the power reflectivity R in this case. As illustrated inFIG. 55, the wavelength band for which the power reflectivity R fallswithin ±2% from the design reflectivity of 8% is from 389 nm to 433 nm,to provide a wavelength bandwidth W_(r) of 44 nm.

Thus, even when the center wavelength λ_(c) is made equal to the designwavelength λ_(t), the wavelength bandwidth W (65 nm) for the coatingfilm 21 is greater than the wavelength bandwidth W_(r) (44 nm) for thecoating film 2 shown in FIG. 2.

Twelfth Preferred Embodiment

FIG. 56 shows the wavelength dependence of the power reflectivity R ofthe coating film 29 in the semiconductor photonic device according tothe twelfth preferred embodiment of the present invention. Thesemiconductor photonic device according to the twelfth preferredembodiment is similar to the semiconductor photonic device shown in FIG.20 according to the fourth preferred embodiment except that the designwavelength λ_(t) is changed from 1550 nm to 650 nm.

For designing the thickness of the coating film 29 having the powerreflectivity R of 8% (R_(t)=8%) when the wavelength λ equals the designwavelength of 650 nm, a point which provides a phase angle θ of 300degrees is selected, for example, in Step s1 so that the reflectionamplitude vector rv is located in the fourth quadrant (or the lowerright quadrant) of the complex plane according to the twelfth preferredembodiment. Then, the values of the real and imaginary parts r_(r) andr_(i) of the complex number inputted as the value of the amplitudereflectivity r are “+0.141421356” and “−0.244948974,” respectively, inStep s2.

When A=1.23, B=2.00, C=2.00 and d₃=10.0 nm are set, the basic amounts ofphase change φ₁ and φ₂ determined in Step s5 are “0.707793” and“2.25201,” respectively. Accordingly, the thicknesses of the second toseventh layer films 23 to 28 determined in Step s6 are 43.78 nm, 176.89nm, 71.19 nm, 287.62 nm, 71.19 nm and 287.62 nm, respectively.

FIG. 56 described above shows the wavelength dependence of the powerreflectivity R of the coating film 29 thus designed. As illustrated inFIG. 56, the power reflectivity R is 8% when the wavelength λ equals thedesign wavelength of 650 nm. A wavelength band for which the powerreflectivity R is approximately equal to the design reflectivity of 8%is wide. The power reflectivity R falls within a range from 6.0% to10.0% for the wavelength λ ranging from 594 nm to 709 nm.

When the allowable reflectivity range is, for example, ±2% from thedesign reflectivity of 8%, the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range is alsofrom 594 nm to 709 nm, to provide a wavelength bandwidth W of 115 nm.The center wavelength λ_(c) of the wavelength band is 652 nm. The valueobtained by dividing the wavelength bandwidth W by the center wavelengthλ_(c) is approximately 0.176, which is greater than 0.06. It may be saidthat the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range is a wide band.

Because the thicknesses of the first to seventh layer films 22 to 28take on the above-mentioned values, the optical thickness t of thecoating film 29 is 1622.10 nm. This value is approximately 9.95 timesthe value t_(r) (163 nm) which is a quarter of the center wavelengthλ_(c). Thus, the coating film 29 is a very thick film. This improvesheat dissipation characteristics at the end surface 1 b of thesemiconductor photonic element 1 to suppress the increase in temperatureof the end surface 1 b.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 8% for the wavelength λ equal to λ_(c) (652 nm)when the coating film 2 having the refractive index n_(f) of “1.3726” asin the ninth preferred embodiment and a thickness d_(f) of five times118.75 nm, i.e. five times λ_(c)/(4n_(f)), is provided on the endsurface 1 b of the semiconductor photonic element 1. FIG. 57 shows thewavelength dependence of the power reflectivity R in this case. Asillustrated in FIG. 57, the wavelength band for which the powerreflectivity R falls within ±2% from the design reflectivity of 8% isfrom 630 nm to 675 nm, to provide a wavelength bandwidth W_(r) of 45 nm.

Thus, the wavelength bandwidth W (115 nm) for the coating film 29 of thetwelfth preferred embodiment is greater than the wavelength bandwidthW_(r) (45 nm) for the coating film 2 shown in FIG. 2.

As mentioned above, because the wavelength λ of light propagatingthrough the active layer 1 a is sometimes varied, it is desirable thatthe center wavelength λ_(c) of the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range be equal toor close to the design wavelength λ_(t). For example, the thicknessd₃=10.0 nm is set, and the basic amounts of phase change φ₁ and φ₂ areset at “0.70747” and “2.25219,” respectively, by adjusting theparameters A to C. The center wavelength λ_(c) takes on a value equal tothe design wavelength of 650 nm when the thickness of the coating film29 is determined using the basic thicknesses d₁ and d₂ obtained bysubstituting the design wavelength of 650 nm for λ in Equations (5a) and(5b). FIG. 58 shows the wavelength dependence of the power reflectivityR in this case.

As illustrated in FIG. 58, the power reflectivity R falls within a rangefrom 6.0% to 10.0% for the wavelength λ ranging from 592 nm to 707 nm.The wavelength band for which the power reflectivity R falls within ±2%from the design reflectivity of 8% is also from 592 nm to 707 nm, toprovide a wavelength bandwidth W of 115 nm. The center wavelength λ_(c)of the wavelength band is 650 nm, which is equal to the designwavelength of 650 nm. The value obtained by dividing the wavelengthbandwidth W (115 nm) by the center wavelength λ_(c) (650 nm) isapproximately 0.177, which is greater than 0.06 and shows that thewavelength band for which the power reflectivity R falls within theallowable reflectivity range is a wide band.

In this case, the thicknesses of the first to seventh layer films 22 to28 of the coating film 29 are 10.00 nm, 43.63 nm, 176.36 nm, 70.94 nm,286.76 nm, 70.94 nm and 286.76 nm, respectively. The optical thickness tof the coating film 29 is 1617.12 nm. This value is approximately 9.92times the value t_(r) (163 nm) which is a quarter of the centerwavelength λ_(c). Thus, the coating film 29 is a very thick film.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 8% for the wavelength λ equal to λ_(c) (650 nm)when the coating film 2 having the above-mentioned refractive indexn_(f) of “1.3726” and a thickness d_(f) of five times 118.39 nm, i.e.five times λ_(c)/(4n_(f)), is provided on the end surface 1 b of thesemiconductor photonic element 1. FIG. 59 shows the wavelengthdependence of the power reflectivity R in this case. As illustrated inFIG. 59, the wavelength band for which the power reflectivity R fallswithin ±2% from the design reflectivity of 8% is from 629 nm to 673 nm,to provide a wavelength bandwidth W_(r) of 44 nm.

Thus, even when the center wavelength λ_(c) is made equal to the designwavelength λ_(t), the wavelength bandwidth W (115 nm) for the coatingfilm 29 is greater than the wavelength bandwidth W_(r) (44 nm) for thecoating film 2 shown in FIG. 2.

Thirteenth Preferred Embodiment

FIG. 60 shows the wavelength dependence of the power reflectivity R ofthe coating film 38 in the semiconductor photonic device according tothe thirteenth preferred embodiment of the present invention. Thesemiconductor photonic device according to the thirteenth preferredembodiment is similar to the semiconductor photonic device shown in FIG.25 according to the fifth preferred embodiment except that the designwavelength λ_(t) is changed from 410 nm to 980 nm.

For designing the thickness of the coating film 38 having the powerreflectivity R of 8% (R_(t)=8%) when the wavelength λ equals the designwavelength of 980 nm, a point which provides a phase angle θ of 15degrees is selected, for example, in Step s1 so that the reflectionamplitude vector rv is located in the first quadrant (or the upper rightquadrant), of the complex plane according to the thirteenth preferredembodiment. Then, the values of the real and imaginary parts r_(r) andr_(i) of the complex number inputted as the value of the amplitudereflectivity r are “+0.27320508” and “+0.07320508,” respectively, inStep s2.

When A=1.96, B=1.31, C=2.02 and D=2.00 are set, the basic amounts ofphase change φ₁ and φ₂ determined in Step s5 are “0.30917” and“1.12523,” respectively. Accordingly, the thicknesses of the first toeighth layer films 30 to 37 determined in Step s6 are 45.95 nm, 232.49nm, 26.49 nm, 134.00 nm, 47.35 nm, 239.54 nm, 46.89 nm and 237.17 nm,respectively.

FIG. 60 described above shows the wavelength dependence of the powerreflectivity R of the coating film 38 thus designed. As illustrated inFIG. 60, the power reflectivity R is 8% when the wavelength λ equals thedesign wavelength of 980 nm. A wavelength band for which the powerreflectivity R is approximately equal to the design reflectivity of 8%is wide. The power reflectivity R falls within a range from 6.0% to10.0% for the wavelength λ ranging from 934 nm to 1129 nm.

When the allowable reflectivity range is, for example, ±2% from thedesign reflectivity of 8%, the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range is alsofrom 934 nm to 1129 nm, to provide a wavelength bandwidth W of 195 nm.The center wavelength λ_(c) of the wavelength band is 1032 nm. The valueobtained by dividing the wavelength bandwidth W by the center wavelengthλ_(c) is approximately 0.189, which is greater than 0.06. It may be saidthat the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range is a wide band.

Because the thicknesses of the first to eighth layer films 30 to 37 takeon the above-mentioned values, the optical thickness t of the coatingfilm 38 is 1590.80 nm. This value is approximately 6.17 times the valuet_(r) (258 nm) which is a quarter of the center wavelength λ_(c). Thus,the coating film 38 is a very thick film. This improves heat dissipationcharacteristics at the end surface 1 b of the semiconductor photonicelement 1 to suppress the increase in temperature of the end surface 1b.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 8% for the wavelength λ equal to λ_(c) (1032 nm)when the coating film 2 having the refractive index n_(f) of “1.3726” asin the ninth preferred embodiment and a thickness d_(f) of five times187.96 nm, i.e. five times λ_(c)/(4n_(f)), is provided on the endsurface 1 b of the semiconductor photonic element 1. FIG. 61 shows thewavelength dependence of the power reflectivity R in this case. Asillustrated in FIG. 61, the wavelength band for which the powerreflectivity R falls within ±2% from the design reflectivity of 8% isfrom 998 nm to 1069 nm, to provide a wavelength bandwidth W_(r) of 71nm.

Thus, the wavelength bandwidth W (195 nm) for the coating film 38 of thethirteenth preferred embodiment is greater than the wavelength bandwidthW_(r) (71 nm) for the coating film 2 shown in FIG. 2.

As mentioned above, because the wavelength λ of light propagatingthrough the active layer 1 a is sometimes varied, it is desirable thatthe center wavelength λ_(c) of the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range be equal toor close to the design wavelength λ_(t). For example, the centerwavelength λ_(c) takes on a value equal to the design wavelength λ_(t)of 980 nm when the thickness of the coating film 38 is determined usingthe basic thicknesses d₁ and d₂ obtained by setting the basic amounts ofphase change φ₁ and φ₂ at “0.30917” and “1.12523,” respectively, in asimilar manner to the above instance and substituting 931 nm, ratherthan the design wavelength λ_(t), for λ in Equations (5a) and (5b). FIG.62 shows the wavelength dependence of the power reflectivity R in thiscase.

As illustrated in FIG. 62, the power reflectivity R falls within a rangefrom 6.0% to 10.0% for the wavelength λ ranging from 887 nm to 1073 nm.The wavelength band for which the power reflectivity R falls within ±2%from the design reflectivity of 8% is also from 887 nm to 1073 nm, toprovide a wavelength bandwidth W of 186 nm. The center wavelength λ_(c)of the wavelength band is 980 nm, which is equal to the designwavelength of 980 nm. The value obtained by dividing the wavelengthbandwidth W (186 nm) by the center wavelength λ_(c) (980 nm) isapproximately 0.190, which is greater than 0.06 and shows that thewavelength band for which the power reflectivity R falls within theallowable reflectivity range is a wide band.

In this case, the thicknesses of the first to eighth layer films 30 to37 of the coating film 38 are 43.65 nm, 220.80 nm, 25.17 nm, 127.30 nm,44.99 nm, 227.56 nm, 44.54 nm and 225.31 nm, respectively. The opticalthickness t of the coating film 38 is 1511.16 nm. This value isapproximately 6.17 times the value t_(r) (245 nm) which is a quarter ofthe center wavelength λ_(c). Thus, the coating film 38 is a very thickfilm.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 8% for the wavelength λ equal to λ_(c) (980 nm)when the coating film 2 having the above-mentioned refractive indexn_(f) of “1.3726” and a thickness d_(f) of five times 178.49 nm, i.e.five times λ_(c)/(4n_(f)), is provided on the end surface 1 b of thesemiconductor photonic element 1. FIG. 63 shows the wavelengthdependence of the power reflectivity R in this case. As illustrated inFIG. 63, the wavelength band for which the power reflectivity R fallswithin ±2% from the design reflectivity of 8% is from 947 nm to 1015 nm,to provide a wavelength bandwidth W_(r) of 68 nm.

Thus, even when the center wavelength λ_(c) is made equal to the designwavelength λ_(t), the wavelength bandwidth W (186 nm) for the coatingfilm 38 is greater than the wavelength bandwidth W_(r) (68 nm) for thecoating film 2 shown in FIG. 2.

Fourteenth Preferred Embodiment

FIG. 64 shows the wavelength dependence of the power reflectivity R ofthe coating film 47 in the semiconductor photonic device according tothe fourteenth preferred embodiment of the present invention. Thesemiconductor photonic device according to the fourteenth preferredembodiment is similar to the semiconductor photonic device shown in FIG.30 according to the sixth preferred embodiment except that the materiallayer for the second layer film 40, the fourth layer film 42, the sixthlayer film 44 and the eighth layer film 46 is changed from the siliconoxide layer to a tantalum oxide layer, that the material layer for thethird layer film 41, the fifth layer film 43 and the seventh layer film45 is changed from the tantalum oxide layer to a silicon oxide layer,and that the design wavelength λ_(t) is changed from 650 nm to 808 nm.Accordingly, the values of the refractive indices n₁ and n₂ are “1.480”and “2.057,” respectively, according to the fourteenth preferredembodiment.

For designing the thickness of the coating film 47 having the powerreflectivity R of 8% (R_(t)=8%) when the wavelength λ equals the designwavelength of 808 nm, a point which provides a phase angle θ of 105degrees is selected, for example, in Step s1 so that the reflectionamplitude vector rv is located in the second quadrant (or the upper leftquadrant) of the complex plane according to the fourteenth preferredembodiment. Then, the values of the real and imaginary parts r_(r) andr_(i) of the complex number inputted as the value of the amplitudereflectivity r are “−0.07320508” and “+0.27320508,” respectively, inStep s2.

When A=2.13, B=2.33, C=2.00, D=2.00 and the thickness d₃=20.0 nm areset, the basic amounts of phase change φ₁ and φ₂ determined in Step s5are “1.94893” and “0.761851,” respectively. Accordingly, the thicknessesof the second to eighth layer films 40 to 46 determined in Step s6 are101.45 nm, 394.57 nm, 110.98 nm, 338.69 nm, 95.26 nm, 338.69 nm and95.26 nm, respectively.

FIG. 64 described above shows the wavelength dependence of the powerreflectivity R of the coating film 47 thus designed. As illustrated inFIG. 64, the power reflectivity R is 8% when the wavelength λ equals thedesign wavelength of 808 nm. A wavelength band for which the powerreflectivity R is approximately equal to the design reflectivity of 8%is wide. The power reflectivity R falls within a range from 6.3% to10.0% for the wavelength λ ranging from 804 nm to 893 nm.

When the allowable reflectivity range is, for example, ±2% from thedesign reflectivity of 8%, the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range is alsofrom 804 nm to 893 nm, to provide a wavelength bandwidth W of 89 nm. Thecenter wavelength λ_(c) of the wavelength band is 849 nm. The valueobtained by dividing the wavelength bandwidth W by the center wavelengthλ_(c) is approximately 0.105, which is greater than 0.06. It may be saidthat the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range is a wide band.

Because the thicknesses of the first to eighth layer films 39 to 46 takeon the above-mentioned values, the optical thickness t of the coatingfilm 47 is 2456.79 nm. This value is approximately 11.59 times the valuet_(r) (212 nm) which is a quarter of the center wavelength λ_(c). Thus,the coating film 47 is a very thick film. This improves heat dissipationcharacteristics at the end surface 1 b of the semiconductor photonicelement 1 to suppress the increase in temperature of the end surface 1b.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 8% for the wavelength λ equal to λ_(c) (849 nm)when the coating film 2 having the refractive index n_(f) of “1.3726” asin the ninth preferred embodiment and a thickness d_(f) of five times154.63 nm, i.e. five times λ_(c)/(4n_(f)), is provided on the endsurface 1 b of the semiconductor photonic element 1. FIG. 65 shows thewavelength dependence of the power reflectivity R in this case. Asillustrated in FIG. 65, the wavelength band for which the powerreflectivity R falls within ±2% from the design reflectivity of 8% isfrom 821 nm to 879 nm, to provide a wavelength bandwidth W_(r) of 58 nm.

Thus, the wavelength bandwidth W (89 nm) for the coating film 47 of thefourteenth preferred embodiment is greater than the wavelength bandwidthW_(r) (58 nm) for the coating film 2 shown in FIG. 2.

As mentioned above, because the wavelength λ of light propagatingthrough the active layer 1 a is sometimes varied, it is desirable thatthe center wavelength λ_(c) of the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range be equal toor close to the design wavelength λ_(t). For example, the thicknessd₃=20.0 nm is set in the above-mentioned instance, and the basic amountsof phase change φ₁ and φ₂ are set at “1.96555” and “0.745004,”respectively, by adjusting the parameters A to D. The center wavelengthλ_(c) takes on a value equal to the design wavelength λ_(t) of 808 nmwhen the thickness of the coating film 47 is determined using the basicthicknesses d₁ and d₂ obtained by substituting 770 nm, rather than thedesign wavelength λ_(t), for λ in Equations (5a) and (5b). FIG. 66 showsthe wavelength dependence of the power reflectivity R in this case.

As illustrated in FIG. 66, the power reflectivity R falls within a rangefrom 6.1% to 10.0% for the wavelength λ ranging from 766 nm to 850 nm.The wavelength band for which the power reflectivity R falls within ±2%from the design reflectivity of 8% is also from 766 nm to 850 nm, toprovide a wavelength bandwidth W of 84 nm. The center wavelength λ_(c)of the wavelength band is 808 nm, which is equal to the designwavelength of 808 nm. The value obtained by dividing the wavelengthbandwidth W (84 nm) by the center wavelength λ_(c) (808 nm) isapproximately 0.104, which is greater than 0.06 and shows that thewavelength band for which the power reflectivity R falls within theallowable reflectivity range is a wide band.

In this case, the thicknesses of the first to eighth layer films 39 to46 of the coating film 47 are 20.00 nm, 94.54 nm, 379.22 nm, 103.42 nm,325.51 nm, 88.77 nm, 325.51 nm and 88.77 nm, respectively. The opticalthickness t of the coating film 47 is 2338.60 nm. This value isapproximately 11.58 times the value t_(r) (202 nm) which is a quarter ofthe center wavelength λ_(c). Thus, the coating film 47 is a very thickfilm.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 8% for the wavelength λ equal to λ_(c) (808 nm)when the coating film 2 having the above-mentioned refractive indexn_(f) of “1.3726” and a thickness d_(f) of five times 147.17 nm, i.e.five times λ_(c)/(4n_(f)), is provided on the end surface 1 b of thesemiconductor photonic element 1. FIG. 67 shows the wavelengthdependence of the power reflectivity R in this case. As illustrated inFIG. 67, the wavelength band for which the power reflectivity R fallswithin ±2% from the design reflectivity of 8% is from 781 nm to 837 nm,to provide a wavelength bandwidth W_(r) of 56 nm.

Thus, even when the center wavelength λ_(c) is made equal to the designwavelength λ_(t), the wavelength bandwidth W (84 nm) for the coatingfilm 47 is greater than the wavelength bandwidth W_(r) (56 nm) for thecoating film 2 shown in FIG. 2.

Fifteenth Preferred Embodiment

FIG. 68 shows the wavelength dependence of the power reflectivity R ofthe coating film 57 in the semiconductor photonic device according tothe fifteenth preferred embodiment of the present invention. Thesemiconductor photonic device according to the fifteenth preferredembodiment is similar to the semiconductor photonic device shown in FIG.35 according to the seventh preferred embodiment except that thematerial layer for the first layer film 48, the third layer film 50, thefifth layer film 52, the seventh layer film 54 and the ninth layer film56 is changed from the alumina layer to a tantalum oxide layer, that thematerial layer for the second layer film 49, the fourth layer film 51,the sixth layer film 53 and the eighth layer film 55 is changed from thetantalum oxide layer to an alumina layer, and that the design wavelengthλ_(t) is changed from 980 nm to 1310 nm. Accordingly, the values of therefractive indices n₁ and n₂ are “1.620” and “2.057,” respectively.

For designing the thickness of the coating film 57 having the powerreflectivity R of 8% (R_(t)=8%) when the wavelength λ equals the designwavelength of 1310 nm, a point which provides a phase angle θ of 195degrees is selected, for example, in Step s1 so that the reflectionamplitude vector rv is located in the third quadrant (or the lower leftquadrant) of the complex plane according to the fifteenth preferredembodiment. Then, the values of the real and imaginary parts r_(r) andr_(i) of the complex number inputted as the value of the amplitudereflectivity r are “−0.27320508” and “−0.07320508,” respectively, inStep s2.

When O=2.05, A=4.20, B=2.00, C=2.00 and D=2.00 are set, the basicamounts of phase change φ₁ and φ₂ determined in Step s5 are “0.410749”and “0.777027,” respectively. Accordingly, the thicknesses of the firstto ninth layer films 48 to 56 determined in Step s6 are 161.45 nm,222.03 nm, 330.78 nm, 105.73 nm, 157.52 nm, 105.73 nm, 157.52 nm, 105.73nm and 157.52 nm, respectively.

FIG. 68 described above shows the wavelength dependence of the powerreflectivity R of the coating film 57 thus designed. As illustrated inFIG. 68, the power reflectivity R is 8% when the wavelength λ equals thedesign wavelength of 1310 nm. A wavelength band for which the powerreflectivity R is approximately equal to the design reflectivity of 8%is wide. The power reflectivity R falls within a range from 6.0% to10.0% for the wavelength λ ranging from 1358 nm to 1626 nm.

When the allowable reflectivity range is, for example, ±2% from thedesign reflectivity of 8%, the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range is alsofrom 1358 nm to 1626 nm, to provide a wavelength bandwidth W of 268 nm.The center wavelength λ_(c) of the wavelength band is 1492 nm. The valueobtained by dividing the wavelength bandwidth W by the center wavelengthλ_(c) is approximately 0.180, which is greater than 0.06. It may be saidthat the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range is a wide band.

Because the thicknesses of the first to ninth layer films 48 to 56 takeon the above-mentioned values, the optical thickness t of the coatingfilm 57 is 2858.11 nm. This value is approximately 7.66 times the valuet_(r) (373 nm) which is a quarter of the center wavelength λ_(c). Thus,the coating film 57 is a very thick film. This improves heat dissipationcharacteristics at the end surface 1 b of the semiconductor photonicelement 1 to suppress the increase in temperature of the end surface 1b.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 8% for the wavelength λ equal to λ_(c) (1492 nm)when the coating film 2 having the refractive index n_(f) of “1.3726” asin the ninth preferred embodiment and a thickness d_(f) of five times271.75 nm, i.e. five times λ_(c)/(4n_(f)), is provided on the endsurface 1 b of the semiconductor photonic element 1. FIG. 69 shows thewavelength dependence of the power reflectivity R in this case. Asillustrated in FIG. 69, the wavelength band for which the powerreflectivity R falls within ±2% from the design reflectivity of 8% isfrom 1442 nm to 1545 nm, to provide a wavelength bandwidth W_(r) of 103nm.

Thus, the wavelength bandwidth W (268 nm) for the coating film 57 of thefifteenth preferred embodiment is greater than the wavelength bandwidthW_(r) (103 nm) for the coating film 2 shown in FIG. 2.

As mentioned above, because the wavelength λ of light propagatingthrough the active layer 1 a is sometimes varied, it is desirable thatthe center wavelength λ_(c) of the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range be equal toor close to the design wavelength λ_(t). For example, the centerwavelength λ_(c) takes on a value equal to the design wavelength λ_(t)of 1310 nm when the thickness of the coating film 57 is determined usingthe basic thicknesses d₁ and d₂ obtained by setting the basic amounts ofphase change φ₁ and φ₂ at “0.410749” and “0.777027,” respectively, in asimilar manner to the above instance and substituting 1150 nm, ratherthan the design wavelength λ_(t), for λ in Equations (5a) and (5b). FIG.70 shows the wavelength dependence of the power reflectivity R in thiscase.

As illustrated in FIG. 70, the power reflectivity R falls within a rangefrom 6.0% to 10.0% for the wavelength λ ranging from 1192 nm to 1427 nm.The wavelength band for which the power reflectivity R falls within ±2%from the design reflectivity of 8% is also from 1192 nm to 1427 nm, toprovide a wavelength bandwidth W of 235 nm. The center wavelength λ_(c)of the wavelength band is 1310 nm, which is equal to the designwavelength of 1310 nm. The value obtained by dividing the wavelengthbandwidth W (235 nm) by the center wavelength λ_(c) (1310 nm) isapproximately 0.179, which is greater than 0.06 and shows that thewavelength band for which the power reflectivity R falls within theallowable reflectivity range is a wide band.

In this case, the thicknesses of the first to ninth layer films 48 to 56of the coating film 57 are 141.73 nm, 194.91 nm, 290.38 nm, 92.81 nm,138.28 nm, 92.81 nm, 138.28 nm, 92.81 nm and 138.28 nm, respectively.The optical thickness t of the coating film 57 is 2508.99 nm. This valueis approximately 7.65 times the value t_(r) (328 nm) which is a quarterof the center wavelength λ_(c). Thus, the coating film 57 is a verythick film.

For the semiconductor photonic device shown in FIG. 2, the powerreflectivity R is 8% for the wavelength λ equal to λ_(c) (1310 nm) whenthe coating film 2 having the above-mentioned refractive index n_(f) of“1.3726” and a thickness d_(f) of five times 238.60 nm, i.e. five timesλ_(c)/(4n_(f)), is provided on the end surface 1 b of the semiconductorphotonic element 1. FIG. 71 shows the wavelength dependence of the powerreflectivity R in this case. As illustrated in FIG. 71, the wavelengthband for which the power reflectivity R falls within ±2% from the designreflectivity of 8% is from 1266 nm to 1357 nm, to provide a wavelengthbandwidth W_(r) of 91 nm.

Thus, even when the center wavelength λ_(c) is made equal to the designwavelength λ_(t), the wavelength bandwidth W (235 nm) for the coatingfilm 57 is greater than the wavelength bandwidth W_(r) (91 nm) for thecoating film 2 shown in FIG. 2.

Sixteenth Preferred Embodiment

FIG. 72 shows the wavelength dependence of the power reflectivity R ofthe coating film 68 in the semiconductor photonic device according tothe sixteenth preferred embodiment of the present invention. Thesemiconductor photonic device according to the sixteenth preferredembodiment is similar to the semiconductor photonic device shown in FIG.39 according to the eighth preferred embodiment except that the materiallayer for the first layer film 58 is changed from the alumina layer toan aluminum nitride layer, that the material layer for the third layerfilm 60, the fifth layer film 62, the seventh layer film 64 and theninth layer film 66 is changed from the aluminum nitride layer to atantalum oxide layer, and that the design wavelength λ_(t) is changedfrom 808 nm to 1550 nm. Accordingly, the values of the refractiveindices n₁ and n₂ are “2.057” and “2.072,” respectively, according tothe sixteenth preferred embodiment.

For designing the thickness of the coating film 68 having the powerreflectivity R of 8% (R_(t)=8%) when the wavelength λ equals the designwavelength of 1550 nm, a point which provides a phase angle θ of 285degrees is selected, for example, in Step s1 so that the reflectionamplitude vector rv is located in the fourth quadrant (or the lowerright quadrant) of the complex plane according to the sixteenthpreferred embodiment. Then, the values of the real and imaginary partsr_(r) and r_(i) of the complex number inputted as the value of theamplitude reflectivity r are “+0.07320508” and “−0.27320508,”respectively, in Step s2.

When A=2.10, B=1.30, C=2.00, D=2.00, E=1.65 and the thickness d₃=20.0 nmare set, the basic amounts of phase change φ₁ and φ₂ determined in Steps5 are “0.722395” and “1.59546,” respectively. Accordingly, thethicknesses of the second to tenth layer films 59 to 67 determined inStep s6 are 558.47 nm, 112.63 nm, 345.72 nm, 173.27 nm, 531.87 nm,173.27 nm, 531.87 nm, 142.95 nm and 438.79 nm, respectively.

FIG. 72 described above shows the wavelength dependence of the powerreflectivity R of the coating film 68 thus designed. As illustrated inFIG. 72, the power reflectivity R is 8% when the wavelength λ equals thedesign wavelength of 1550 nm. A wavelength band for which the powerreflectivity R is approximately equal to the design reflectivity of 8%is wide. The power reflectivity R falls within a range from 7.6% to10.0% for the wavelength λ ranging from 1534 nm to 1659 nm.

When the allowable reflectivity range is, for example, ±2% from thedesign reflectivity of 8%, the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range is alsofrom 1534 nm to 1659 nm, to provide a wavelength bandwidth W of 125 nm.The center wavelength λ_(c) of the wavelength band is 1597 nm. The valueobtained by dividing the wavelength bandwidth W by the center wavelengthλ_(c) is approximately 0.078, which is greater than 0.06. It may be saidthat the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range is a wide band.

Because the thicknesses of the first to tenth layer films 58 to 67 takeon the above-mentioned values, the optical thickness t of the coatingfilm 68 is 4841.95 nm. This value is approximately 12.14 times the valuet_(r) (399 nm) which is a quarter of the center wavelength λ_(c). Thus,the coating film 68 is a very thick film. This improves heat dissipationcharacteristics at the end surface 1 b of the semiconductor photonicelement 1 to suppress the increase in temperature of the end surface 1b.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 8% for the wavelength λ equal to λ_(c) (1597 nm)when the coating film 2 having the refractive index n_(f) of “1.3726” asin the ninth preferred embodiment and a thickness d_(f) of five times290.87 nm, i.e. five times λ_(c)/(4n_(f)), is provided on the endsurface 1 b of the semiconductor photonic element 1. FIG. 73 shows thewavelength dependence of the power reflectivity R in this case. Asillustrated in FIG. 73, the wavelength band for which the powerreflectivity R falls within ±2% from the design reflectivity of 8% isfrom 1544 nm to 1654 nm, to provide a wavelength bandwidth W_(r) of 110nm.

Thus, the wavelength bandwidth W (125 nm) for the coating film 68 of thesixteenth preferred embodiment is greater than the wavelength bandwidthW_(r) (110 nm) for the coating film 2 shown in FIG. 2.

As mentioned above, because the wavelength λ of light propagatingthrough the active layer 1 a is sometimes varied, it is desirable thatthe center wavelength λ_(c) of the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range be equal toor close to the design wavelength λ_(t). For example, when the thicknessd₃=20.0 nm, A=2.10, B=1.30, C=2.00, D=2.00 and E=1.65 are set, the basicamounts of phase change φ₁ and φ₂ are “0.723268” and “1.59370,”respectively. The center wavelength λ_(c) takes on a value equal to thedesign wavelength of 1550 nm when the thickness of the coating film 68is determined using the basic thicknesses d₁ and d₂ obtained bysubstituting the above-mentioned values of the basic amounts of phasechange φ₁ and φ₂ into Equations (5a) and (5b) and substituting 1505 nm,rather than the design wavelength λ_(t), for λ in Equations (5a) and(5b). FIG. 74 shows the wavelength dependence of the power reflectivityR in this case.

As illustrated in FIG. 74, the power reflectivity R falls within a rangefrom 7.6% to 10.0% for the wavelength λ ranging from 1489 nm to 1610 nm.The wavelength band for which the power reflectivity R falls within ±2%from the design reflectivity of 8% is also from 1489 nm to 1610 nm, toprovide a wavelength bandwidth W of 121 nm. The center wavelength λ_(c)of the wavelength band is 1550 nm, which is equal to the designwavelength of 1550 nm. The value obtained by dividing the wavelengthbandwidth W (121 nm) by the center wavelength λ_(c) (1550 nm) isapproximately 0.078, which is greater than 0.06 and shows that thewavelength band for which the power reflectivity R falls within theallowable reflectivity range is a wide band.

In this case, the thicknesses of the first to tenth layer films 58 to 67of the coating film 68 are 20.00 nm, 541.65 nm, 109.49 nm, 335.31 nm,168.44 nm, 515.86 nm, 168.44 nm, 515.86 nm, 138.97 nm and 425.59 nm,respectively. The optical thickness t of the coating film 68 is 4700.20nm. This value is approximately 12.11 times the value t_(r) (388 nm)which is a quarter of the center wavelength λ_(c). Thus, the coatingfilm 68 is a very thick film.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 8% for the wavelength λ equal to λ_(c) (1550 nm)when the coating film 2 having the above-mentioned refractive indexn_(f) of “1.3726” and a thickness d_(f) of five times 282.31 nm, i.e.five times λ_(c)/(4n_(f)), is provided on the end surface 1 b of thesemiconductor photonic element 1. FIG. 75 shows the wavelengthdependence of the power reflectivity R in this case. As illustrated inFIG. 75, the wavelength band for which the power reflectivity R fallswithin ±2% from the design reflectivity of 8% is from 1498 nm to 1606nm, to provide a wavelength bandwidth W_(r) of 108 nm.

Thus, even when the center wavelength λ_(c) is made equal to the designwavelength λ₁, the wavelength bandwidth W (121 nm) for the coating film68 is greater than the wavelength bandwidth W_(r) (108 nm) for thecoating film 2 shown in FIG. 2.

Seventeenth Preferred Embodiment

FIG. 76 shows the wavelength dependence of the power reflectivity R ofthe coating film 21 in the semiconductor photonic device according tothe seventeenth preferred embodiment of the present invention. Thesemiconductor photonic device according to the seventeenth preferredembodiment is similar to the semiconductor photonic device shown in FIG.15 according to the third preferred embodiment except that the materiallayer for the first layer film 14, the third layer film 16, the fifthlayer film 18 and the seventh layer film 20 is changed from the aluminalayer to a tantalum oxide layer, that the material layer for the secondlayer film 15, the fourth layer film 17 and the sixth layer film 19 ischanged from the tantalum oxide layer to a silicon oxide layer, and thatthe design wavelength λ_(t) is changed from 1310 nm to 980 nm.Accordingly, the values of the refractive indices n₁ and n₂ are “1.480”and “2.057,” respectively.

For designing the thickness of the coating film 21 having a powerreflectivity R of 12% (R_(t)=12%) when the wavelength λ equals thedesign wavelength of 980 nm, a point which provides a phase angle θ of75 degrees is selected, for example, in Step s1 so that the reflectionamplitude vector rv is located in the first quadrant (or the upper rightquadrant) of the complex plane according to the seventeenth preferredembodiment. Then, because the magnitude of the complex number at theselected point is 0.346410161, the values of the real and imaginaryparts r_(r) and r_(i) of the complex number inputted as the value of theamplitude reflectivity r are “+0.089657547” and “+0.334606521,”respectively, in Step s2.

When O=2.56, A=2.95, B=2.00 and C=2.00 are set, the basic amounts ofphase change φ₁ and φ₂ determined in Step s5 are “1.43423” and“0.68016,” respectively. Accordingly, the thicknesses of the first toseventh layer films 14 to 20 determined in Step s6 are 132.03 nm, 445.89nm, 152.14 nm, 302.30 nm, 103.15 nm, 302.30 nm and 103.15 nm,respectively.

FIG. 76 described above shows the wavelength dependence of the powerreflectivity R of the coating film 21 thus designed. As illustrated inFIG. 76, the power reflectivity R is 12% when the wavelength λ equalsthe design wavelength of 980 nm. A wavelength band for which the powerreflectivity R is approximately equal to the design reflectivity of 12%is wide. The power reflectivity R falls within a range from 11.3% to14.0% for the wavelength λ ranging from 938 nm to 1087 nm.

When the allowable reflectivity range is, for example, ±2% from thedesign reflectivity of 12%, the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range is alsofrom 938 nm to 1087 nm, to provide a wavelength bandwidth W of 149 nm.The center wavelength λ_(c) of the wavelength band is 1013 nm. The valueobtained by dividing the wavelength bandwidth W by the center wavelengthλ_(c) is approximately 0.147, which is greater than 0.06. It may be saidthat the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range is a wide band.

Because the thicknesses of the first to seventh layer films 14 to 20take on the above-mentioned values, the optical thickness t of thecoating film 21 is 2563.62 nm. This value is approximately 10.13 timesthe value t_(r) (253 nm) which is a quarter of the center wavelengthλ_(c). Thus, the coating film 21 is a very thick film. This improvesheat dissipation characteristics at the end surface 1 b of thesemiconductor photonic element 1 to suppress the increase in temperatureof the end surface 1 b.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 12% for the wavelength λ equal to λ_(c) (1013nm) when the coating film 2 having a refractive index n_(f) of “1.27902”and a thickness d_(f) of five times 199.00 nm, i.e. five timesλ_(c)/(4n_(f)), is provided on the end surface 1 b of the semiconductorphotonic element 1. FIG. 77 shows the wavelength dependence of the powerreflectivity R in this case. As illustrated in FIG. 77, the wavelengthband for which the power reflectivity R falls within ±2% from the designreflectivity of 12% is from 975 nm to 1054 nm, to provide a wavelengthbandwidth W_(r) of 79 nm.

The above-mentioned value “1.27902” of the refractive index n_(f) isobtained by substituting R_(t)=0.12 and n_(c)=3.37 into Equation (8).

Thus, the wavelength bandwidth W (149 nm) for the coating film 21 of theseventeenth preferred embodiment is greater than the wavelengthbandwidth W_(r) (79 nm) for the coating film 2 shown in FIG. 2.

As mentioned above, because the wavelength λ of light propagatingthrough the active layer 1 a is sometimes varied, it is desirable thatthe center wavelength λ_(c) of the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range be equal toor close to the design wavelength λ_(t). For example, the centerwavelength λ_(c) takes on a value equal to the design wavelength λ_(t)of 980 nm when the thickness of the coating film 21 is determined usingthe basic thicknesses d₁ and d₂ obtained by setting the basic amounts ofphase change φ₁ and φ₂ at “1.43423” and “0.68016,” respectively, in asimilar manner to the above instance and substituting 948 nm, ratherthan the design wavelength λ_(t), for λ in Equations (5a) and (5b). FIG.78 shows the wavelength dependence of the power reflectivity R in thiscase.

As illustrated in FIG. 78, the power reflectivity R falls within a rangefrom 11.3% to 14.0% for the wavelength λ ranging from 907 nm to 1052 nm.The wavelength band for which the power reflectivity R falls within ±2%from the design reflectivity of 12% is also from 907 nm to 1052 nm, toprovide a wavelength bandwidth W of 145 nm. The center wavelength λ_(c)of the wavelength band is 980 nm, which is equal to the designwavelength of 980 nm. The value obtained by dividing the wavelengthbandwidth W (145 nm) by the center wavelength λ_(c) (980 nm) isapproximately 0.148, which is greater than 0.06 and shows that thewavelength band for which the power reflectivity R falls within theallowable reflectivity range is a wide band.

In this case, the thicknesses of the first to seventh layer films 14 to20 of the coating film 21 are 127.72 nm, 431.33 nm, 147.17 nm, 292.43nm, 99.78 nm, 292.43 nm and 99.78 nm, respectively. The opticalthickness t of the coating film 21 is 2479.91 nm. This value isapproximately 10.12 times the value t_(r) (245 nm) which is a quarter ofthe center wavelength λ_(c). Thus, the coating film 21 is a very thickfilm.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 12% for the wavelength λ equal to λ_(c) (980 nm)when the coating film 2 having the above-mentioned refractive indexn_(f) of “1.27902” and a thickness d_(f) of five times 191.55 nm, i.e.five times λ_(c)/(4n_(f)), is provided on the end surface 1 b of thesemiconductor photonic element 1. FIG. 79 shows the wavelengthdependence of the power reflectivity R in this case. As illustrated inFIG. 79, the wavelength band for which the power reflectivity R fallswithin ±2% from the design reflectivity of 12% is from 943 nm to 1020nm, to provide a wavelength bandwidth W_(r) of 77 nm.

Thus, even when the center wavelength λ_(c) is made equal to the designwavelength λ_(t), the wavelength bandwidth W (145 nm) for the coatingfilm 21 is greater than the wavelength bandwidth W_(r) (77 nm) for thecoating film 2 shown in FIG. 2.

Eighteenth Preferred Embodiment

FIG. 80 shows the wavelength dependence of the power reflectivity R ofthe coating film 21 such that the power reflectivity R is 16% when thewavelength λ equals the design wavelength of 980 nm in the semiconductorphotonic device of the above-mentioned seventeenth preferred embodimentof the present invention.

For designing the thickness of the coating film 21 having the powerreflectivity R of 16% when the wavelength λ equals the design wavelengthof 980 nm, a point which provides a phase angle θ of 165 degrees isselected, for example, in Step s1 so that the reflection amplitudevector rv is located in the second quadrant (or the upper left quadrant)of the complex plane according to the above-mentioned seventeenthpreferred embodiment. Then, because the magnitude of the complex numberat the selected point is 0.4, the values of the real and imaginary partsr_(r) and r_(i) of the complex number inputted as the value of theamplitude reflectivity r are “−0.38637033” and “+0.103527618,”respectively, in Step s2.

When O=2.50, A=3.75, B=3.37 and C=1.80 are set, the basic amounts ofphase change φ₁ and φ₂ determined in Step s5 are “0.651305” and“0.381901,” respectively. Accordingly, the thicknesses of the first toseventh layer films 14 to 20 determined in Step s6 are 72.39 nm, 257.40nm, 108.59 nm, 231.31 nm, 97.59 nm, 123.55 nm and 52.12 nm,respectively.

FIG. 80 described above shows the wavelength dependence of the powerreflectivity R of the coating film 21 thus designed. As illustrated inFIG. 80, the power reflectivity R is 16% when the wavelength λ equalsthe design wavelength of 980 nm. A wavelength band for which the powerreflectivity R is approximately equal to the design reflectivity of 16%is wide. The power reflectivity R falls within a range from 15.0% to18.0% for the wavelength λ ranging from 945 nm to 1219 nm.

When the allowable reflectivity range is, for example, ±2% from thedesign reflectivity of 16%, the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range is alsofrom 945 nm to 1219 nm, to provide a wavelength bandwidth W of 274 nm.The center wavelength λ_(c) of the wavelength band is 1082 nm. The valueobtained by dividing the wavelength bandwidth W by the center wavelengthλ_(c) is approximately 0.253, which is greater than 0.06. It may be saidthat the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range is a wide band.

Because the thicknesses of the first to seventh layer films 14 to 20take on the above-mentioned values, the optical thickness t of thecoating film 21 is 1586.37 nm. This value is approximately 5.85 timesthe value t_(r) (271 nm) which is a quarter of the center wavelengthλ_(c). Thus, the coating film 21 is a very thick film. This improvesheat dissipation characteristics at the end surface 1 b of thesemiconductor photonic element 1 to suppress the increase in temperatureof the end surface 1 b.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 16% for the wavelength λ equal to λ_(c) (1082nm) when the coating film 2 having a refractive index n_(f) of“1.20178.” and a thickness d_(f) of five times 225.08 nm, i.e. fivetimes λ_(c)/(4n_(f)), is provided on the end surface 1 b of thesemiconductor photonic element 1. FIG. 81 shows the wavelengthdependence of the power reflectivity R in this case. As illustrated inFIG. 81, the wavelength band for which the power reflectivity R fallswithin ±2% from the design reflectivity of 16% is from 1034 nm to 1134nm, to provide a wavelength bandwidth W_(r) of 100 nm.

The above-mentioned value “1.20178” of the refractive index n_(f) isobtained by substituting R_(t)=0.16 and n_(c)=3.37 into Equation (8).

Thus, the wavelength bandwidth W (274 nm) for the coating film 21 of theeighteenth preferred embodiment of the present invention is greater thanthe wavelength bandwidth W_(r) (100 nm) for the coating film 2 shown inFIG. 2.

As mentioned above, because the wavelength λ of light propagatingthrough the active layer 1 a is sometimes varied, it is desirable thatthe center wavelength λ_(c) of the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range be equal toor close to the design wavelength λ_(t). For example, the centerwavelength λ_(c) takes on a value equal to the design wavelength λ_(t)of 980 nm when the thickness of the coating film 21 is determined usingthe basic thicknesses d₁ and d₂ obtained by setting the basic amounts ofphase change φ₁ and φ₂ at “0.651305” and “0.381901,” respectively, in asimilar manner to the above instance and substituting 887 nm, ratherthan the design wavelength λ_(t), for λ in Equations (5a) and (5b). FIG.82 shows the wavelength dependence of the power reflectivity R in thiscase.

As illustrated in FIG. 82, the power reflectivity R falls within a rangefrom 15.0% to 18.0% for the wavelength λ ranging from 856 nm to 1103 nm.The wavelength band for which the power reflectivity R falls within ±2%from the design reflectivity of 16% is from 856 nm to 1103 nm, toprovide a wavelength bandwidth W of 247 nm. The center wavelength λ_(c)of the wavelength band is 980 nm, which is equal to the designwavelength of 980 nm. The value obtained by dividing the wavelengthbandwidth W (247 nm) by the center wavelength λ_(c) (980 nm) isapproximately 0.252, which is greater than 0.06 and shows that thewavelength band for which the power reflectivity R falls within theallowable reflectivity range is a wide band.

In this case, the thicknesses of the first to seventh layer films 14 to20 of the coating film 21 are 65.52 nm, 232.97 nm, 98.29 nm, 209.36 nm,88.33 nm, 111.83 nm and 47.18 nm, respectively. The optical thickness tof the coating film 21 is 1435.86 nm. This value is approximately 5.86times the value t_(r) (245 nm) which is a quarter of the centerwavelength λ_(c). Thus, the coating film 21 is a very thick film.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 16% for the wavelength λ equal to λ_(c) (980 nm)when the coating film 2 having the above-mentioned refractive indexn_(f) of “1.20178” and a thickness d_(f) of five times 203.86 nm, i.e.five times λ_(c)/(4n_(f)), is provided on the end surface 1 b of thesemiconductor photonic element 1. FIG. 83 shows the wavelengthdependence of the power reflectivity R in this case. As illustrated inFIG. 83, the wavelength band for which the power reflectivity R fallswithin ±2% from the design reflectivity of 16% is from 937 nm to 1027nm, to provide a wavelength bandwidth W_(r) of 90 nm.

Thus, even when the center wavelength λ_(c) is made equal to the designwavelength λ_(t), the wavelength bandwidth W (247 nm) for the coatingfilm 21 is greater than the wavelength bandwidth W_(r) (90 nm) for thecoating film 2 shown in FIG. 2.

Nineteenth Preferred Embodiment

FIG. 84 shows the wavelength dependence of the power reflectivity R ofthe coating film 21 such that the power reflectivity R is 20% when thewavelength λ equals the design wavelength of 980 nm in the semiconductorphotonic device of the above-mentioned seventeenth preferred embodimentof the present invention.

For designing the thickness of the coating film 21 having the powerreflectivity R of 20% when the wavelength λ equals the design wavelengthof 980 nm, a point which provides a phase angle θ of 255 degrees isselected, for example, in Step s1 so that the reflection amplitudevector rv is located in the third quadrant (or the lower left quadrant)of the complex plane according to the above-mentioned seventeenthpreferred embodiment. Then, because the magnitude of the complex numberat the selected point is 0.447213595, the values of the real andimaginary parts r_(r) and r_(i) of the complex number inputted as thevalue of the amplitude reflectivity r are “−0.115747395” and“−0.431975161,” respectively, in Step s2.

When O=2.00, A=2.00, B=2.71 and C=1.02 are set, the basic amounts ofphase change φ₁ and φ₂ determined in Step s5 are “1.01212” and“0.703719,” respectively. Accordingly, the thicknesses of the first toseventh layer films 14 to 20 determined in Step s6 are 106.72 nm, 213.33nm, 106.72 nm, 289.06 nm, 144.60 nm, 108.80 nm and 54.43 nm,respectively.

FIG. 84 described above shows the wavelength dependence of the powerreflectivity R of the coating film 21 thus designed. As illustrated inFIG. 84, the power reflectivity R is 20% when the wavelength λ equalsthe design wavelength of 980 nm. A wavelength band for which the powerreflectivity R is approximately equal to the design reflectivity of 20%is wide. The power reflectivity R falls within a range from 18.0% to22.0% for the wavelength λ ranging from 911 nm to 1365 nm.

When the allowable reflectivity range is, for example, ±2% from thedesign reflectivity of 20%, the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range is alsofrom 911 nm to 1365 nm, to provide a wavelength bandwidth W of 454 nm.The center wavelength λ_(c) of the wavelength band is 1138 nm. The valueobtained by dividing the wavelength bandwidth W by the center wavelengthλ_(c) is approximately 0.399, which is greater than 0.06. It may be saidthat the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range is a wide band.

Because the thicknesses of the first to seventh layer films 14 to 20take on the above-mentioned values, the optical thickness t of thecoating film 21 is 1753.01 nm. This value is approximately 6.15 timesthe value t_(r) (285 nm) which is a quarter of the center wavelengthλ_(c). Thus, the coating film 21 is a very thick film. This improvesheat dissipation characteristics at the end surface 1 b of thesemiconductor photonic element 1 to suppress the increase in temperatureof the end surface 1 b.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 20% for the wavelength λ equal to λ_(c) (1138nm) when the coating film 2 having a refractive index n_(f) of “1.13456”and a thickness d_(f) of five times 250.76 nm, i.e. five timesλ_(c)/(4n_(f)), is provided on the end surface 1 b of the semiconductorphotonic element 1. FIG. 85 shows the wavelength dependence of the powerreflectivity R in this case. As illustrated in FIG. 85, the wavelengthband for which the power reflectivity R falls within ±2% from the designreflectivity of 20% is from 1076 nm to 1207 nm, to provide a wavelengthbandwidth W_(r) of 131 nm.

The above-mentioned value “1.13456” of the refractive index n_(f) isobtained by substituting R_(t)=0.20 and n_(c)=3.37 into Equation (8).

Thus, the wavelength bandwidth W (454 nm) for the coating film 21 of thenineteenth preferred embodiment of the present invention is greater thanthe wavelength bandwidth W_(r) (131 nm) for the coating film 2 shown inFIG. 2.

As mentioned above, because the wavelength λ of light propagatingthrough the active layer 1 a is sometimes varied, it is desirable thatthe center wavelength λ_(c) of the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range be equal toor close to the design wavelength λ_(t). For example, the centerwavelength λ_(c) takes on a value equal to the design wavelength λ_(t)of 980 nm when the thickness of the coating film 21 is determined usingthe basic thicknesses d₁ and d₂ obtained by setting the basic amounts ofphase change φ₁ and φ₂ at “1.01212” and “0.703719,” respectively, in asimilar manner to the above instance and substituting 844 nm, ratherthan the design wavelength λ₁, for λ in Equations (5a) and (5b). FIG. 86shows the wavelength dependence of the power reflectivity R in thiscase.

As illustrated in FIG. 86, the power reflectivity R falls within a rangefrom 18.0% to 22.0% for the wavelength λ ranging from 784 nm to 1176 nm.The wavelength band for which the power reflectivity R falls within ±2%from the design reflectivity of 20% is also from 784 nm to 1176 nm, toprovide a wavelength bandwidth W of 392 nm. The center wavelength λ_(c)of the wavelength band is 980 nm, which is equal to the designwavelength of 980 nm. The value obtained by dividing the wavelengthbandwidth W (392 nm) by the center wavelength λ_(c) (980 nm) isapproximately 0.400, which is greater than 0.06 and shows that thewavelength band for which the power reflectivity R falls within theallowable reflectivity range is a wide band.

In this case, the thicknesses of the first to seventh layer films 14 to20 of the coating film 21 are 91.91 nm, 183.72 nm, 91.91 nm, 248.94 nm,124.54 nm, 93.70 nm and 46.87 nm, respectively. The optical thickness tof the coating film 21 is 1509.72 nm. This value is approximately 6.16times the value t_(r) (245 nm) which is a quarter of the centerwavelength λ_(c). Thus, the coating film 21 is a very thick film.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 20% for the wavelength λ equal to λ_(c) (980 nm)when the coating film 2 having the above-mentioned refractive indexn_(f) of “1.13456” and a thickness d_(f) of five times 215.94 nm, i.e.five times λ_(c)/(4n_(f)), is provided on the end surface 1 b of thesemiconductor photonic element 1. FIG. 87 shows the wavelengthdependence of the power reflectivity R in this case. As illustrated inFIG. 87, the wavelength band for which the power reflectivity R fallswithin ±2% from the design reflectivity of 20% is from 927 nm to 1040nm, to provide a wavelength bandwidth W_(r) of 113 nm.

Thus, even when the center wavelength λ_(c) is made equal to the designwavelength λ_(t), the wavelength bandwidth W (392 nm) for the coatingfilm 21 is greater than the wavelength bandwidth W_(r) (113 nm) for thecoating film 2 shown in FIG. 2.

Twentieth Preferred Embodiment

FIG. 88 shows the wavelength dependence of the power reflectivity R ofthe coating film 21 such that the power reflectivity R is 24% when thewavelength λ equals the design wavelength of 980 nm in the semiconductorphotonic device of the above-mentioned seventeenth preferred embodimentof the present invention.

For designing the thickness of the coating film 21 having the powerreflectivity R of 24% when the wavelength λ equals the design wavelengthof 980 nm, a point which provides a phase angle θ of 345 degrees isselected, for example, in Step s1 so that the reflection amplitudevector rv is located in the fourth quadrant (or the lower rightquadrant) of the complex plane according to the above-mentionedseventeenth preferred embodiment. Then, because the magnitude of thecomplex number at the selected point is 0.489897948, the values of thereal and imaginary parts r_(r) and r_(i) of the complex number inputtedas the value of the amplitude reflectivity r are “+0.47320508” and“−0.126794919,” respectively, in Step s2.

When O=1.95, A=1.85, B=0.10 and C=1.96 are set, the basic amounts ofphase change φ₁ and φ₂ determined in Step s5 are “1.40351” and“0.680892,” respectively. Accordingly, the thicknesses of the first toseventh layer films 14 to 20 determined in Step s6 are 100.68 nm, 273.64nm, 95.51 nm, 14.79 nm, 5.16 nm, 289.91 nm and 101.19 nm, respectively.

FIG. 88 described above shows the wavelength dependence of the powerreflectivity R of the coating film 21 thus designed. As illustrated inFIG. 88, the power reflectivity R is 24% when the wavelength λ equalsthe design wavelength of 980 nm. A wavelength band for which the powerreflectivity R is approximately equal to the design reflectivity of 24%is wide. The power reflectivity R falls within a range from 22.0% to24.0% for the wavelength λ ranging from 961 nm to 1153 nm.

When the allowable reflectivity range is, for example, ±2% from thedesign reflectivity of 24%, the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range is alsofrom 961 nm to 1153 nm, to provide a wavelength bandwidth W of 192 nm.The center wavelength λ_(c) of the wavelength band is 1057 nm. The valueobtained by dividing the wavelength bandwidth W by the center wavelengthλ_(c) is approximately 0.182, which is greater than 0.06. It may be saidthat the wavelength band for which the power reflectivity R falls withinthe allowable reflectivity range is a wide band.

Because the thicknesses of the first to seventh layer films 14 to 20take on the above-mentioned values, the optical thickness t of thecoating film 21 is 1478.27 nm. This value is approximately 5.60 timesthe value t_(r) (264 nm) which is a quarter of the center wavelengthλ_(c). Thus, the coating film 21 is a very thick film. This improvesheat dissipation characteristics at the end surface 1 b of thesemiconductor photonic element 1 to suppress the increase in temperatureof the end surface 1 b.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 24% for the wavelength λ equal to λ_(c) (1057nm) when the coating film 2 having a refractive index n_(f) of “1.07415”and a thickness d_(f) of five times 246.01 nm, i.e. five timesλ_(c)/(4n_(f)), is provided on the end surface 1 b of the semiconductorphotonic element 1. FIG. 89 shows the wavelength dependence of the powerreflectivity R in this case. As illustrated in FIG. 89, the wavelengthband for which the power reflectivity R falls within ±2% from the designreflectivity of 24% is from 978 nm to 1150 nm, to provide a wavelengthbandwidth W_(r) of 172 nm.

The above-mentioned value “1.07415” of the refractive index n_(f) isobtained by substituting R_(t)=0.24 and n_(c)=3.37 into Equation (8).

Thus, the wavelength bandwidth W (192 nm) for the coating film 21 of thetwentieth preferred embodiment of the present invention is greater thanthe wavelength bandwidth W_(r) (172 nm) for the coating film 2 shown inFIG. 2.

As mentioned above, because the wavelength λ of light propagatingthrough the active layer 1 a is sometimes varied, it is desirable thatthe center wavelength λ_(c) of the wavelength band for which the powerreflectivity R falls within the allowable reflectivity range be equal toor close to the design wavelength λ_(t). For example, the centerwavelength λ_(c) takes on a value approximately equal to the designwavelength λ_(t) of 980 nm when the thickness of the coating film 21 isdetermined using the basic thicknesses d₁ and d₂ obtained by setting thebasic amounts of phase change φ₁ and φ₂ at “1.40351” and “0.680892,”respectively, in a similar manner to the above instance and substituting909 nm, rather than the design wavelength λ_(t), for λ in Equations (5a)and (5b). FIG. 90 shows the wavelength dependence of the powerreflectivity R in this case.

As illustrated in FIG. 90, the power reflectivity R falls within a rangefrom 22.0% to 26.0% for the wavelength λ ranging from 891 nm to 1070 nm.The wavelength band for which the power reflectivity R falls within ±2%from the design reflectivity of 24% is also from 891 nm to 1070 nm, toprovide a wavelength bandwidth W of 179 nm. The center wavelength λ_(c)of the wavelength band is 981 nm, which is approximately equal to thedesign wavelength of 980 nm. The value obtained by dividing thewavelength bandwidth W (179 nm) by the center wavelength λ_(c) (981 nm)is approximately 0.182, which is greater than 0.06 and shows that thewavelength band for which the power reflectivity R falls within theallowable reflectivity range is a wide band.

In this case, the thicknesses of the first to seventh layer films 14 to20 of the coating film 21 are 93.38 nm, 253.81 nm, 88.59 nm, 13.72 nm,4.79 nm, 268.90 nm and 93.86 nm, respectively. The optical thickness tof the coating film 21 is 1371.15 nm. This value is approximately 5.60times the value t_(r) (245 nm) which is a quarter of the centerwavelength λ_(c). Thus, the coating film 21 is a very thick film.

The semiconductor photonic device shown in FIG. 2 will be considered forcomparison. For the semiconductor photonic device shown in FIG. 2, thepower reflectivity R is 24% for the wavelength λ equal to λ_(c) (981 nm)when the coating film 2 having the above-mentioned refractive indexn_(f) of “1.07415” and a thickness d_(f) of five times 228.32 nm, i.e.five times λ_(c)/(4n_(f)), is provided on the end surface 1 b of thesemiconductor photonic element 1. FIG. 91 shows the wavelengthdependence of the power reflectivity R in this case. As illustrated inFIG. 91, the wavelength band for which the power reflectivity R fallswithin ±2% from the design reflectivity of 24% is from 907 nm to 1066nm, to provide a wavelength bandwidth W_(r) of 159 nm.

Thus, even when the center wavelength λ_(c) is made close to the designwavelength λ_(t), the wavelength bandwidth W (179 nm) for the coatingfilm 21 is greater than the wavelength bandwidth W_(r) (159 nm) for thecoating film 2 shown in FIG. 2.

Although the center wavelength λ_(c) is close to the design wavelengthλ_(t) in the above instance, the center wavelength λ_(c) may be madeexactly equal to the design wavelength λ_(t) by adjusting the values ofthe parameters A, B, C and O or the value substituted for λ in Equations(5a) and (5b).

The conditions and results according to the first to twentieth preferredembodiments described hereinabove are listed in FIGS. 92 and 93.

As described hereinabove, the first to twentieth preferred embodimentsof the present invention employ an imaginary number as the value of theamplitude reflectivity r of the coating film provided on the end surface1 b of the semiconductor photonic element 1, to make it possible todesign the thickness of the coating film having the predetermined powerreflectivity R in consideration for more complex numbers having the sameamplitude as the value of the amplitude reflectivity r than realnumbers. This improves the design flexibility of the thickness of thecoating film to make the coating film having a desired characteristiceasy to design.

Additionally, the first to twentieth preferred embodiments determine thethicknesses of the respective layers included in the coating film sothat the center wavelength λ_(c) of the wavelength band for which thepower reflectivity R of the coating film falls within the allowablereflectivity range is equal to the design wavelength λ_(t). Thisprovides the coating film whose power reflectivity R is varied slightlyeven if the wavelength λ in the actual device is changed from the designwavelength λ_(t).

In the semiconductor photonic device according to the first to twentiethpreferred embodiments, the power reflectivity R of the coating film isvaried slightly even if the wavelength λ of light propagating throughthe active layer 1 a is varied, because of the wide wavelength band forwhich the power reflectivity R of the coating film falls within theallowable reflectivity range. This achieves the provision of thesemiconductor photonic device having characteristics less susceptible tothe wavelength dependence of the power reflectivity R of the coatingfilm.

The first to twentieth preferred embodiments reliably suppress thevariation in the power reflectivity R of the coating film even if thewavelength λ is varied because the allowable reflectivity range is setat ±2% from the median value thereof (the design reflectivity R_(t)).

The coating films described above have the six-layer structure, theseven-layer structure, the eight-layer structure, the nine-layerstructure, and the ten-layer structure in the first to twentiethpreferred embodiments. The present invention, however, is not limited tothese coating films, but is applicable to coating films having otherlayer structures.

The values of the parameters A, B, C, D, E and O are merely examples.The parameters A, B, C, D, E and O may take on other values to producesimilar effects.

The design wavelength λ_(t) is 410 nm, 650 nm, 808 nm, 980 nm, 1310 nmand 1550 nm in the first to twentieth preferred embodiments describedabove. The present invention, however, is not limited to this, but isapplicable to other values of the wavelength.

Although only up to three material layers for the coating film aredescribed, the present invention is applicable to four or more materiallayers for the coating film.

While the invention has been described in detail, the foregoingdescription is in all aspects illustrative and not restrictive. It isunderstood that numerous other modifications and variations can bedevised without departing from the scope of the invention.

1. A method of designing the thickness of a coating film including a plurality of layers and provided on an end surface of a semiconductor photonic element including an active layer through which light propagates, said method comprising the steps of: (a) selecting an imaginary number as a value of an amplitude reflectivity of said coating film; and (b) determining the thickness of each of said plurality of layers of said coating film so that the value of said amplitude reflectivity of said coating film is equal to said imaginary number selected in said step (a).
 2. The method according to claim 1, wherein the thickness of each of said plurality of layers of said coating film is determined in said step (b) so that a center wavelength of a wavelength band for which a power reflectivity of said coating film falls within a predetermined range when the wavelength of light propagating through said active layer is hypothetically varied is equal to a design value of said wavelength.
 3. A semiconductor photonic device comprising: a semiconductor photonic element including an active layer through which light propagates; and a coating film including a plurality of layers and provided on an end surface of said semiconductor photonic element, said coating film having an amplitude reflectivity taking on a value set at an imaginary value.
 4. The semiconductor photonic device according to claim 3, wherein the width of a first wavelength band for which a power reflectivity of said coating film falls within a predetermined range when the wavelength of light propagating through said active layer is hypothetically varied is greater than the width of a second wavelength band for which a power reflectivity of a single layer film obtained when said single layer film is provided on said end surface of said active layer falls within said predetermined range when said wavelength is hypothetically varied, said single layer film having a refractive index satisfying R _(t)=((n _(c) −n _(f) ²)/(n _(c) +n _(f) ²))² where n_(f) is the refractive index of said single layer film, n_(c) is an effective refractive index of said semiconductor photonic element, and R_(t) is a median value of said predetermined range, said single layer film having a thickness 5/(4n_(f)) times the center wavelength of said first wavelength band.
 5. The semiconductor photonic device according to claim 3, wherein the value obtained by dividing the width of a wavelength band for which a power reflectivity of said coating film falls within a predetermined range when the wavelength of light propagating through said active layer is hypothetically varied by the center wavelength of said wavelength band is greater than 0.06.
 6. The semiconductor photonic device according to claim 4, wherein the sum of the products of the thicknesses and refractive indices of the respective layers of said coating film is greater than 3λ_(c)/4 where λ_(c) is said center wavelength of said first wavelength band.
 7. The semiconductor photonic device according to claim 5, wherein the sum of the products of the thicknesses and refractive indices of the respective layers of said coating film is greater than 3λ_(c)/4 where λ_(c) is said center wavelength of said wavelength band.
 8. The semiconductor photonic device according to claim 4, wherein said predetermined range is ±2% from the median value thereof.
 9. The semiconductor photonic device according to claim 5, wherein said predetermined range is ±2% from the median value thereof.
 10. The semiconductor photonic device according to claim 3, wherein a power reflectivity of said coating film is less than a power reflectivity at said end surface defined by the effective refractive index of said semiconductor photonic element and a refractive index in a free space contacting said end surface when said coating film is absent.
 11. The semiconductor photonic device according to claim 3, wherein said coating film includes two material layers, and the number of layers of said coating film is selected from the group consisting of six, seven, eight and nine.
 12. The semiconductor photonic device according to claim 3, wherein said coating film includes three material layers, and the number of layers of said coating film is selected from the group consisting of seven, eight and ten.
 13. The semiconductor photonic device according to claim 3, wherein said coating film includes at least two selected from the group consisting of a silicon oxide layer, a tantalum oxide layer, an alumina layer and an aluminum nitride layer. 